Mercurial > repos > public > sbplib
changeset 425:e56dbd9e4196 feature/grids
Merge feature/beams
| author | Jonatan Werpers <jonatan@werpers.com> |
|---|---|
| date | Tue, 07 Feb 2017 16:09:02 +0100 |
| parents | a2cb0d4f4a02 (current diff) 37649ef3fc6e (diff) |
| children | eca4ca84cf0a eef74cd9b49c |
| files | +sbp/+implementations/d4_compatible_halfvariable_2.m +sbp/+implementations/d4_compatible_halfvariable_4.m +sbp/+implementations/d4_compatible_halfvariable_6.m +sbp/D4CompatibleVariable.m |
| diffstat | 83 files changed, 2600 insertions(+), 996 deletions(-) [+] |
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diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/fromMatrix.m --- a/+blockmatrix/fromMatrix.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+blockmatrix/fromMatrix.m Tue Feb 07 16:09:02 2017 +0100 @@ -13,7 +13,7 @@ for j = 1:length(d2) Asub = A(I:(I + d1(i)-1), J:(J + d2(j)-1)); if nnz(Asub) == 0 - bm{i,j} = []; + bm{i,j} = sparse(d1(i), d2(j)); else bm{i,j} = Asub; end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/isBlockmatrixTest.m --- a/+blockmatrix/isBlockmatrixTest.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+blockmatrix/isBlockmatrixTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -52,6 +52,10 @@ }, true % Empty blocks allowed. }, + { + blockmatrix.zero({[1 2 3],[2 3]}), + true % A zero block matrix is a block matrix + }, }; for i = 1:length(cases)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/multiply.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/multiply.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,17 @@ +function C = multiply(A, B) + [n_a, m_a] = size(A); + [n_b, m_b] = size(B); + + assert(m_a == n_b, 'Dimensions do not agree.') + + C = cell(n_a, m_b); + + for i = 1:n_a + for j = 1:m_b + C{i,j} = sparse(size(A{i,1},1), size(B{1,j},2)); + for k = 1:n_b + C{i,j} = C{i,j} + A{i,k}*B{k,j}; + end + end + end +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/multiplyTest.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+blockmatrix/multiplyTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,57 @@ +function tests = multiplyTest() + tests = functiontests(localfunctions); +end + + +function testMultiply(testCase) + a11 = [ + 0.8147 0.1270; + 0.9058 0.9134; + ]; + a12 = [ + 0.6324 0.2785 0.9575; + 0.0975 0.5469 0.9649; + ]; + a21 = [ + 0.1576 0.9706; + ]; + a22 = [ + 0.9572 0.4854 0.8003; + ]; + A = { + a11 a12; + a21 a22; + }; + + b11 = [ + 0.1419 0.9157 0.9595; + 0.4218 0.7922 0.6557; + ]; + b12 = [ + 0.0357 0.9340; + 0.8491 0.6787; + ]; + b21 = [ + 0.7577 0.6555 0.0318; + 0.7431 0.1712 0.2769; + 0.3922 0.7060 0.0462; + ]; + b22 = [ + 0.0971 0.3171; + 0.8235 0.9502; + 0.6948 0.0344; + ]; + + B = { + b11 b12; + b21 b22; + }; + + + C = { + a11*b11 + a12*b21, a11*b12 + a12*b22; + a21*b11 + a22*b21, a21*b12 + a22*b22; + }; + + testCase.verifyEqual(blockmatrix.multiply(A,B), C); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/toMatrix.m --- a/+blockmatrix/toMatrix.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+blockmatrix/toMatrix.m Tue Feb 07 16:09:02 2017 +0100 @@ -20,6 +20,7 @@ if isempty(bm{i,j}) continue end + % TODO: If this ever fails for large matrices. Try cell2mat instead. A(n_ind(i)+1:n_ind(i+1),m_ind(j)+1:m_ind(j+1)) = bm{i,j}; end end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +blockmatrix/zero.m --- a/+blockmatrix/zero.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+blockmatrix/zero.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,9 @@ % Creates a block matrix according to the division with zeros everywhere. function bm = zero(div) + if ~blockmatrix.isDivision(div); + error('div is not a valid division'); + end + n = div{1}; m = div{2};
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +draw/prompt_bezier.m --- a/+draw/prompt_bezier.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+draw/prompt_bezier.m Tue Feb 07 16:09:02 2017 +0100 @@ -22,8 +22,8 @@ p.Color = Color.yellow; p.MarkerSize = 16; - C = grid.Curve.bezier(a,c1,c2,b); - fprintf('C = grid.Curve.bezier([%.3g; %.3g],[%.3g; %.3g],[%.3g; %.3g],[%.3g; %.3g]);\n',a,c1,c2,b) + C = parametrization.Curve.bezier(a,c1,c2,b); + fprintf('C = parametrization.Curve.bezier([%.3g; %.3g],[%.3g; %.3g],[%.3g; %.3g],[%.3g; %.3g]);\n',a,c1,c2,b) h = C.plot(); uistack(h,'bottom');
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/Curvilinear.m --- a/+grid/Curvilinear.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/Curvilinear.m Tue Feb 07 16:09:02 2017 +0100 @@ -7,7 +7,8 @@ methods % Creates a curvilinear grid. % Ex: grid.Curvilinear(mapping, xi, eta, ...) - % mapping -- either a matrix or a cell array with physical coordinates. + % mapping -- either a function handle, a matrix or a cell array with physical coordinates. + % A function handle should be a vector valued function of the coordinate mapping. % A matrix should be a grid function (N*D x 1 vector) or a N x D % A cell array should be a 1 x D cell array with either N x 1 vectors % or matrices of the same dimesions as the logical grid. @@ -31,7 +32,7 @@ elseif isnumeric(mapping) obj.coords = matrixMappingToCoords(mapping, N, D); else - error('grid:Curvilinear:Curvilinear','mapping must be a matrix or a cell array.'); + error('grid:Curvilinear:Curvilinear','mapping must be a function handle, a matrix or a cell array.'); end end @@ -66,12 +67,12 @@ % Restricts the grid function gf on obj to the subgrid g. function gf = restrictFunc(obj, gf, g) - gf = obj.logic.restrictFunc(gf, g.baseGrid()); + gf = obj.logic.restrictFunc(gf, g.logic); end % Projects the grid function gf on obj to the grid g. function gf = projectFunc(obj, gf, g) - gf = obj.logic.projectFunc(gf,g.baseGrid()); + gf = obj.logic.projectFunc(gf,g.logic); end function h = scaling(obj)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/equidistantCurvilinear.m --- a/+grid/equidistantCurvilinear.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/equidistantCurvilinear.m Tue Feb 07 16:09:02 2017 +0100 @@ -31,5 +31,5 @@ end g = grid.Curvilinear(mapping, X{:}); - g.h = h; + g.logic.h = h; end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/equidistantTest.m --- a/+grid/equidistantTest.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/equidistantTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -47,4 +47,4 @@ g = grid.equidistant(in{i}{:}); testCase.verifyEqual(g.points(),out{i}); end -end \ No newline at end of file +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/evalOn.m --- a/+grid/evalOn.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/evalOn.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,4 +1,4 @@ -% Takes a funciton and evaluates it on a grid to return a grid function in the +% Takes a function and evaluates it on a grid to return a grid function in the % form of a (n*k)x1 vector, where n is the number of grid points and k is the % number of components of the function. % g -- Grid to evaluate on. @@ -16,15 +16,18 @@ end % func should now be a function_handle - % Get coordinates and convert to cell array for easier use as a parameter - x = g.points(); - X = {}; - for i = 1:size(x, 2) - X{i} = x(:,i); + if g.D ~= nargin(func) + g.D + nargin(func) + error('grid:evalOn:WrongNumberOfInputs', 'The number of inputs of the function must match the dimension of the domain.') end + + % Get coordinates and convert to cell array for easier use as a parameter + x = num2cell(g.points()); + % Find the number of components - x0 = num2cell(x(1,:)); + x0 = x(1,:); f0 = func(x0{:}); k = length(f0); @@ -32,8 +35,11 @@ error('grid:evalOn:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') end - gf = func(X{:}); - if k > 1 % Reorder so that componets sits together. - gf = reshape(reshape(gf, [g.N, k])', [g.N*k, 1]); + gf = zeros(g.N*k, 1); + % keyboard + for i = 1:g.N + % (1 + (i-1)*k):(i*k) + % func(x{i,:}) + gf((1 + (i-1)*k):(i*k)) = func(x{i,:}); end end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/evalOnScalar.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOnScalar.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,41 @@ +% WHAT KIND OF A FUNCTION NAME IS THIS?! +% This functions send matrixa arguments to func unlike grid.evalOn() +% Takes a funciton and evaluates it on a grid to return a grid function in the +% form of a (n*k)x1 vector, where n is the number of grid points and k is the +% number of components of the function. +% g -- Grid to evaluate on. +% func -- Function to evaluate. May be a function handle or a constant. If +% it is a vector value it has to be provided as a column vector, +function gf = evalOn(g, func) + if ~isa(func, 'function_handle') + % We should have a constant. + if size(func,2) ~= 1 + error('grid:evalOnScalar:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = repmat(func,[g.N, 1]); + return + end + % func should now be a function_handle + + % Get coordinates and convert to cell array for easier use as a parameter + x = g.points(); + X = {}; + for i = 1:size(x, 2) + X{i} = x(:,i); + end + + % Find the number of components + x0 = num2cell(x(1,:)); + f0 = func(x0{:}); + k = length(f0); + + if size(f0,2) ~= 1 + error('grid:evalOnScalar:VectorValuedWrongDim', 'A vector valued function must be given as a column vector') + end + + gf = func(X{:}); + if k > 1 % Reorder so that componets sits together. + gf = reshape(reshape(gf, [g.N, k])', [g.N*k, 1]); + end +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/evalOnScalarTest.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/evalOnScalarTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,118 @@ +function tests = evalOnScalarTest() + tests = functiontests(localfunctions); +end + +function testInputConstant(testCase) + in = { + 0, + 47, + 1, + [1; 2], + }; + + out = { + [0; 0; 0], + [47; 47; 47], + [1; 1; 1], + [1; 2; 1; 2; 1; 2], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +function testInputScalarFunction1d(testCase) + in = { + @(x)1+x*0, + @(x)x, + @(x)x.*x, + }; + + out = { + [1; 1; 1], + [0; 1; 2], + [0; 1; 4], + }; + + g = getTestGrid('1d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g,in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + +function testInputScalarFunction2d(testCase) + in = { + @(x,y)1+x*0, + @(x,y)x-y, + @(x,y)x./(1+y), + }; + + out = { + [1; 1; 1; 1; 1; 1; 1; 1; 1], + [0; -1; -2; 1; 0; -1; 2; 1; 0], + [0; 0; 0; 1; 1/2; 1/3; 2; 1; 2/3], + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + gf = grid.evalOnScalar(g, in{i}); + testCase.verifyEqual(gf, out{i}); + end +end + + +function testInputVectorFunction(testCase) + g = getTestGrid('1d'); + in = @(x)[x; -2*x]; + out = [0; 0; 1; -2; 2; -4]; + + gf = grid.evalOnScalar(g,in); + testCase.verifyEqual(gf, out); + + g = getTestGrid('2d'); + in = @(x,y)[x.^2; -2*y]; + out = [ + 0; 0; + 0; -2; + 0; -4; + 1; 0; + 1; -2; + 1; -4; + 4; 0; + 4; -2; + 4; -4; + ]; + + gf = grid.evalOnScalar(g,in); + testCase.verifyEqual(gf, out); +end + + +function testInputErrorVectorValued(testCase) + in = { + [1,2,3], + @(x,y)[x,-y]; + }; + + g = getTestGrid('2d'); + + for i = 1:length(in) + testCase.verifyError(@()grid.evalOnScalar(g, in{i}),'grid:evalOnScalar:VectorValuedWrongDim',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + +function g = getTestGrid(d) + switch d + case '1d' + g = grid.equidistant(3,{0,2}); + case '2d' + g = grid.equidistant([3,3],{0,2},{0,2}); + end +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/evalOnTest.m --- a/+grid/evalOnTest.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/evalOnTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -25,6 +25,21 @@ end end +% evalOn should give and error if the number of inputs to func is not the same as +% the number of dimensions of the grid. +function testNumberOfInputs(testCase) + cases = { + {getTestGrid('1d'), @(x,y)x-y}, + {getTestGrid('2d'), @(x)x }, + } + + for i = 1:length(cases) + g = cases{i}{1}; + f = cases{i}{2}; + testCase.verifyError(@()grid.evalOn(g, f),'grid:evalOn:WrongNumberOfInputs',sprintf('in(%d) = %s',i,toString(f))); + end +end + function testInputScalarFunction1d(testCase) in = { @(x)1+x*0,
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/funcToComponents.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToComponents.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,8 @@ +% funcToComponents converts a grid function to a N x k matrix, where +% k is the number of vector components of the gridfunction and N is the +% number of points in the grid. +% +% Takes a grid function and and a grid. +function F = funcToComponents(g, gf); + F = reshapeRowMaj(gf, [g.N, length(gf)/g.N]); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/funcToComponentsTest.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/funcToComponentsTest.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,23 @@ +function tests = funcToComponentsTest() + tests = functiontests(localfunctions); +end + + +function testScalarGf(testCase) + g = getTestGrid(); + gf_in = [1; 2; 3]; + + testCase.verifyEqual(grid.funcToComponents(g, gf_in), gf_in); +end + +function testVectorGf(testCase) + g = getTestGrid(); + gf_in = [1; 2; 3; 4; 5; 6]; + out = [1 2; 3 4; 5 6]; + + testCase.verifyEqual(grid.funcToComponents(g, gf_in), out); +end + +function g = getTestGrid() + g = grid.equidistant(3,{0,2}); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/funcToMatrix.m --- a/+grid/funcToMatrix.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/funcToMatrix.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ % Converts a gridfunction to a matrix % Takes a grid function and and a structured grid. function F = funcToMatrix(g, gf) - reshapeRowMaj(gf, g.size()); + F = reshapeRowMaj(gf, g.size()); end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +grid/funcToPlotMatrix.m --- a/+grid/funcToPlotMatrix.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+grid/funcToPlotMatrix.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ % Converts a gridfunction to a plot matrix % Takes a grid function and and a structured grid. function F = funcToPlotMatrix(g, gf) - reshapeToPlotMatrix(gf, g.size()); + F = reshapeToPlotMatrix(gf, g.size()); end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +multiblock/DiffOp.m --- a/+multiblock/DiffOp.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+multiblock/DiffOp.m Tue Feb 07 16:09:02 2017 +0100 @@ -37,6 +37,9 @@ for i = 1:nBlocks h = getHand(i); p = getParam(i); + if ~iscell(p) + p = {p}; + end obj.diffOps{i} = h(grid.grids{i}, order, p{:}); end @@ -50,18 +53,20 @@ % Build the differentiation matrix - D = cell(nBlocks, nBlocks); + obj.blockmatrixDiv = {grid.Ns, grid.Ns}; + D = blockmatrix.zero(obj.blockmatrixDiv); for i = 1:nBlocks D{i,i} = obj.diffOps{i}.D; end for i = 1:nBlocks - for j = i:nBlocks + for j = 1:nBlocks intf = grid.connections{i,j}; if isempty(intf) continue end + [ii, ij] = obj.diffOps{i}.interface(intf{1}, obj.diffOps{j}, intf{2}); D{i,i} = D{i,i} + ii; D{i,j} = D{i,j} + ij; @@ -73,7 +78,6 @@ end obj.D = blockmatrix.toMatrix(D); - obj.blockmatrixDiv = blockmatrix.getDivision(D); function [getHand, getParam] = parseInput(doHand, grid, doParam) if ~isa(grid, 'multiblock.Grid') @@ -90,11 +94,23 @@ if isempty(doParam) getParam = @(i){}; - elseif iscell(doParam) && length(doParam) == grid.nBlocks() + return + end + + if ~iscell(doParam) + getParam = @(i)doParam; + return + end + + % doParam is a non-empty cell-array + + if length(doParam) == grid.nBlocks() && all(cellfun(@iscell, doParam)) + % doParam is a cell-array of cell-arrays getParam = @(i)doParam{i}; - else - getParam = @(i)doParam; + return end + + getParam = @(i)doParam; end end @@ -104,29 +120,59 @@ ops = sparse2cell(op, obj.NNN); end - % Creates the closere and penalty matrix for a given boundary condition, - % boundary -- the name of the boundary on the form [id][name] where + function op = getBoundaryOperator(obj, op, boundary) + if iscell(boundary) + localOpName = [op '_' boundary{2}]; + blockId = boundary{1}; + localOp = obj.diffOps{blockId}.(localOpName); + + div = {obj.blockmatrixDiv{1}, size(localOp,2)}; + blockOp = blockmatrix.zero(div); + blockOp{blockId,1} = localOp; + op = blockmatrix.toMatrix(blockOp); + return + else + % Boundary är en sträng med en boundary group i. + end + end + + % Creates the closure and penalty matrix for a given boundary condition, + % boundary -- the name of the boundary on the form {id,name} where % id is the number of a block and name is the name of a - % boundary of that block example: 1s or 3w - function [closure, penalty] = boundary_condition(obj,boundary,type,data) - I = boundary(1) - name = boundary(2:end); + % boundary of that block example: {1,'s'} or {3,'w'} + function [closure, penalty] = boundary_condition(obj, boundary, type) + I = boundary{1}; + name = boundary{2}; % Get the closure and penaly matrices - [blockClosure, blockPenalty] = obj.diffOps{I}.boundary_condition(name, type, data); + [blockClosure, blockPenalty] = obj.diffOps{I}.boundary_condition(name, type); % Expand to matrix for full domain. div = obj.blockmatrixDiv; - closure = blockmatrix.zero(div); - closure{I,I} = blockClosure; + if ~iscell(blockClosure) + temp = blockmatrix.zero(div); + temp{I,I} = blockClosure; + closure = blockmatrix.toMatrix(temp); + else + for i = 1:length(blockClosure) + temp = blockmatrix.zero(div); + temp{I,I} = blockClosure{i}; + closure{i} = blockmatrix.toMatrix(temp); + end + end - div{2} = 1; % Penalty is a column vector - penalty = blockmatrix.zero(div); - penalty{I} = blockPenalty; - - % Convert to matrix - closure = blockmatrix.toMatrix(closure); - penalty = blockmatrix.toMatrix(closure); + div{2} = size(blockPenalty, 2); % Penalty is a column vector + if ~iscell(blockPenalty) + p = blockmatrix.zero(div); + p{I} = blockPenalty; + penalty = blockmatrix.toMatrix(p); + else + for i = 1:length(blockPenalty) + p = blockmatrix.zero(div); + p{I} = blockPenalty{i}; + penalty{i} = blockmatrix.toMatrix(p); + end + end end function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +multiblock/Grid.m --- a/+multiblock/Grid.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+multiblock/Grid.m Tue Feb 07 16:09:02 2017 +0100 @@ -34,11 +34,20 @@ n = length(obj.grids); end - % n returns the number of points in the grid + % N returns the number of points in the grid function o = N(obj) o = obj.nPoints; end + % Ns returns the number of points in each sub grid as a vector + function o = Ns(obj) + ns = zeros(1,obj.nBlocks); + for i = 1:obj.nBlocks; + ns(i) = obj.grids{i}.N(); + end + o = ns; + end + function n = nBlocks(obj) n = length(obj.grids); end @@ -62,9 +71,9 @@ nBlocks = length(obj.grids); % Collect number of points in each block - N = cell(1,nBlocks); + N = zeros(1,nBlocks); for i = 1:nBlocks - N{i} = obj.grids{i}.N(); + N(i) = obj.grids{i}.N(); end gfs = mat2cell(gf, N, 1);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +multiblock/stitchSchemes.m --- a/+multiblock/stitchSchemes.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+multiblock/stitchSchemes.m Tue Feb 07 16:09:02 2017 +0100 @@ -11,24 +11,24 @@ % Output parameters are cell arrays and cell matrices. % % Ex: [schms, D, H] = stitchSchemes(schmHand, order, schmParam, blocks, ms, conn, bound) -function [schms, D, H] = stitchSchemes(schmHand, order, schmParam, blocks, ms, conn, bound) +function [schms, D, H] = stitchSchemes(schmHand, order, schmParam, grids, conn, bound) default_arg('schmParam',[]); - n_blocks = numel(blocks); + n_blocks = numel(grids); % Creating Schemes for i = 1:n_blocks if isempty(schmParam); - schms{i} = schmHand(ms{i},blocks{i},order,[]); + schms{i} = schmHand(grids{i},order,[]); elseif ~iscell(schmParam) param = schmParam(i); - schms{i} = schmHand(ms{i},blocks{i},order,param); + schms{i} = schmHand(grids{i},order,param); else param = schmParam{i}; if iscell(param) - schms{i} = schmHand(ms{i},blocks{i},order,param{:}); + schms{i} = schmHand(grids{i},order,param{:}); else - schms{i} = schmHand(ms{i},blocks{i},order,param); + schms{i} = schmHand(grids{i},order,param); end end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/animate.m --- a/+noname/animate.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+noname/animate.m Tue Feb 07 16:09:02 2017 +0100 @@ -5,10 +5,14 @@ % noname.animate(discr,1, [tstart tend],'my_mov', opt) function hand = animate(discretization, time_modifier, Tend, dirname, opt) - default_arg('time_modifier',1); + default_arg('time_modifier', 1); default_arg('Tend', Inf); - default_arg('dirname',''); - default_arg('opt', []); + default_arg('dirname', ''); + + optDefault.plotType = 'animation'; + optDefault.time = []; + + default_struct('opt', optDefault); if time_modifier < 0 @@ -34,7 +38,7 @@ fprintf('m : %d\n',size(discretization)); - ts = discretization.getTimestepper(opt); + ts = discretization.getTimestepper(opt.time); if numel(Tend) == 2 Tstart = Tend(1); @@ -50,7 +54,7 @@ Tstart = start_solution.t; end - [update, figure_handle] = discretization.setupPlot('animation'); + [update, figure_handle] = discretization.setupPlot(opt.plotType); if makemovies save_frame = anim.setup_fig_mov(figure_handle,dirname); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/calcSol.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+noname/calcSol.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,10 @@ +% Calculates the solution of a discr at a given time using aligned timesteps. +% Returns the solution as a grid function as defined in +grid +function gf = calculateSolution(discr, T, tsOpt) + k_max = discr.getTimestep(tsOpt); + [k,N] = alignedTimestep(k_max,T); + tsOpt.k = k; + ts = discr.getTimestepper(tsOpt); + + gf = ts.stepN(N-ts.n); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/calculateSolution.m --- a/+noname/calculateSolution.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+noname/calculateSolution.m Tue Feb 07 16:09:02 2017 +0100 @@ -13,7 +13,7 @@ fprintf('File ''%s'' already exist.',filename); do_append = yesnoQuestion('Do you want to append to it?'); if ~do_append - fprintf('Exiting...\n'); + fprintf('Exiting. No Solutions calculated.\n'); return end end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/convergence.m --- a/+noname/convergence.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+noname/convergence.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ % Reference is either a key or a function handle -function [q, e, h, runtime] = convergence(filename, errorFunc, reference, method, order, m, T) +function [q, e, h, runtime] = convergence(filename, errorFunc, reference, name, order, m, T) default_arg('errorFunc', @scheme.error1d); sf = SolutionFile(filename); @@ -7,7 +7,7 @@ % Generate convergence, error, and efficiency plots for each search key with more than one entry. for i = 1:length(m) - key.method = method; + key.name = name; key.order = order; key.m = m(i); key.T = T; @@ -30,18 +30,11 @@ % Get the reference solution vector if isa(reference,'function_handle'); - x = v_repr.x; + x = v_repr.grid.points(); v_ref = reference(x,T); else % Downsample the reference solution - x = v_repr.x; - x_ref = reference.x; - - [~,I] = ismember(x,x_ref,'rows'); - if any(I == 0) - error('Solution and reference solution seem to be on different grids.'); - end - v_ref = reference.v(I); + v_ref = reference.grid.restrictFunc(reference.v, v_repr.grid); end e = errorFunc(discr,v, v_ref);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/plotSolutions.m --- a/+noname/plotSolutions.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+noname/plotSolutions.m Tue Feb 07 16:09:02 2017 +0100 @@ -10,7 +10,7 @@ key = sf.keys{i}; entry = sf.get(key); - method = key.method; + name = key.name; order = key.order; m = key.m; T = key.T; @@ -23,11 +23,11 @@ update(repr); if save_figures - figname = sprintf('%s_%s_o%d_m%d_T%d',figename_prefix,method,order,m,i); + figname = sprintf('%s_%s_o%d_m%d_T%d',figename_prefix,name,order,m,i); fprintf('Saving figure to ''%s''\n',figname); saveeps(hand,figname); end end -end \ No newline at end of file +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +noname/testCfl.m --- a/+noname/testCfl.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+noname/testCfl.m Tue Feb 07 16:09:02 2017 +0100 @@ -27,29 +27,37 @@ end end + if silentFlag + rsInterval = util.ReplaceableString(''); + end + % Use bisection to find sharp estimate while( (alpha0(2)-alpha0(1))/alpha0(1) > tol) alpha = mean(alpha0); if ~silentFlag fprintf('[%.3e,%.3e]: ', alpha0(1), alpha0(2)); + else + rsInterval.update('[%.3e,%.3e]: ', alpha0(1), alpha0(2)); end [ok, n_step, maxVal] = testAlpha(alpha); if ok alpha0(1) = alpha; + stability = 'STABLE'; else alpha0(2) = alpha; + stability = 'UNSTABLE'; end if ~silentFlag - fprintf('a = %.3e, n_step=%d max = %.2e\n', alpha, n_step, maxVal); + fprintf('a = %.3e, n_step=%d %8s max = %.2e\n', alpha, n_step, stability, maxVal); end end if silentFlag - fprintf('Last calculated: a = %.3e, n_step=%d max = %.2e\n', alpha, n_step, maxVal); + rsInterval = util.ReplaceableString(''); end fprintf('T = %-3d dof = %-4d order = %d: clf = %.4e\n',T, discr.size(), discr.order, alpha0(1));
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/Curve.m --- a/+parametrization/Curve.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/Curve.m Tue Feb 07 16:09:02 2017 +0100 @@ -24,7 +24,7 @@ if ~isempty(transformation) transformation.base_g = g; transformation.base_gp = gp; - [g,gp] = grid.Curve.transform_g(g,gp,transformation); + [g,gp] = parametrization.Curve.transform_g(g,gp,transformation); end obj.g = g; @@ -68,17 +68,21 @@ end function h= show(obj,name) + default_arg('name', '') p = obj.g(1/2); n = obj.normal(1/2); p = p + n*0.1; % Add arrow - h = text(p(1),p(2),name); - h.HorizontalAlignment = 'center'; - h.VerticalAlignment = 'middle'; - obj.plot(); + if ~isempty(name) + h = text(p(1),p(2),name); + h.HorizontalAlignment = 'center'; + h.VerticalAlignment = 'middle'; + end + + h = obj.plot(); end % Shows curve with name and arrow for direction. @@ -106,7 +110,7 @@ % New function and derivative g_new = @(t)obj.g(arcPar(t)); gp_new = @(t) normalize(obj.gp(arcPar(t))); - curve = grid.Curve(g_new,gp_new); + curve = parametrization.Curve(g_new,gp_new); end @@ -131,7 +135,7 @@ function D = reverse(C) % g = C.g; % gp = C.gp; - % D = grid.Curve(@(t)g(1-t),@(t)-gp(1-t)); + % D = parametrization.Curve(@(t)g(1-t),@(t)-gp(1-t)); D = C.transform([],[],-1); end @@ -157,7 +161,7 @@ transformation.flip = flip*flip_old; end - D = grid.Curve(g,gp,transformation); + D = parametrization.Curve(g,gp,transformation); end @@ -171,7 +175,7 @@ % v(2,:) = x(2,:)+a(2); % end - % D = grid.Curve(@g_fun,gp); + % D = parametrization.Curve(@g_fun,gp); D = C.transform([],a); end @@ -206,7 +210,7 @@ % v = A*gp(t); % end - % D = grid.Curve(@g_fun,@gp_fun); + % D = parametrization.Curve(@g_fun,@gp_fun); % g = A(g-a)+a = Ag - Aa + a; b = - A*a + a; @@ -237,7 +241,7 @@ % v = A*gp(t); % end - % D = grid.Curve(@g_fun,@gp_fun); + % D = parametrization.Curve(@g_fun,@gp_fun); % g = A(g-a)+a = Ag - Aa + a; @@ -266,9 +270,13 @@ v(1,:) = p1(1) + t.*(p2(1)-p1(1)); v(2,:) = p1(2) + t.*(p2(2)-p1(2)); end - g = @g_fun; - obj = grid.Curve(g); + function v = g_fun_deriv(t) + v(1,:) = t.*0 + (p2(1)-p1(1)); + v(2,:) = t.*0 + (p2(2)-p1(2)); + end + + obj = parametrization.Curve(@g_fun, @g_fun_deriv); end function obj = circle(c,r,phi) @@ -288,7 +296,7 @@ v(2,:) = (phi(2)-phi(1))*r*cos(w); end - obj = grid.Curve(@g_fun,@g_fun_deriv); + obj = parametrization.Curve(@g_fun,@g_fun_deriv); end function obj = bezier(p0, p1, p2, p3) @@ -302,7 +310,7 @@ v(2,:) = 3*(1-t).^2*(p1(2)-p0(2)) + 6*(1-t).*t*(p2(2)-p1(2)) + 3*t.^2*(p3(2)-p2(2)); end - obj = grid.Curve(@g_fun,@g_fun_deriv); + obj = parametrization.Curve(@g_fun,@g_fun_deriv); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/Ti.m --- a/+parametrization/Ti.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/Ti.m Tue Feb 07 16:09:02 2017 +0100 @@ -36,6 +36,8 @@ obj.S = @S_fun; end + % Does this funciton make sense? + % Should it always be eval? function [X,Y] = map(obj,u,v) default_arg('v',u); @@ -65,6 +67,19 @@ end end + % Evaluate S for each pair of u and v, + % Return same shape as u + function [x, y] = eval(obj, u, v) + x = zeros(size(u)); + y = zeros(size(u)); + + for i = 1:numel(u) + p = obj.S(u(i), v(i)); + x(i) = p(1,:); + y(i) = p(2,:); + end + end + function h = plot(obj,nu,nv) S = obj.S; @@ -119,7 +134,7 @@ new_gs{i} = gs{i}.translate(a); end - ti = grid.Ti(new_gs{:}); + ti = parametrization.Ti(new_gs{:}); end % Mirrors the Ti so that the resulting Ti is still left handed. @@ -134,7 +149,7 @@ new_gs{2} = gs{4}.mirror(a,b).reverse(); new_gs{4} = gs{2}.mirror(a,b).reverse(); - ti = grid.Ti(new_gs{:}); + ti = parametrization.Ti(new_gs{:}); end function ti = rotate(obj,a,rad) @@ -144,7 +159,7 @@ new_gs{i} = gs{i}.rotate(a,rad); end - ti = grid.Ti(new_gs{:}); + ti = parametrization.Ti(new_gs{:}); end function ti = rotate_edges(obj,n); @@ -153,18 +168,35 @@ new_i = mod(i - n,4); new_gs{new_i+1} = obj.gs{i+1}; end - ti = grid.Ti(new_gs{:}); + ti = parametrization.Ti(new_gs{:}); end end methods(Static) function obj = points(p1, p2, p3, p4) - g1 = grid.Curve.line(p1,p2); - g2 = grid.Curve.line(p2,p3); - g3 = grid.Curve.line(p3,p4); - g4 = grid.Curve.line(p4,p1); + g1 = parametrization.Curve.line(p1,p2); + g2 = parametrization.Curve.line(p2,p3); + g3 = parametrization.Curve.line(p3,p4); + g4 = parametrization.Curve.line(p4,p1); + + obj = parametrization.Ti(g1,g2,g3,g4); + end - obj = grid.Ti(g1,g2,g3,g4); + % Like the constructor but allows inputing line curves as 2-cell arrays: + % example: parametrization.Ti.linesAndCurves(g1, g2, {a, b} g4) + function obj = linesAndCurves(C1, C2, C3, C4) + C = {C1, C2, C3, C4}; + c = cell(1,4); + + for i = 1:4 + if ~iscell(C{i}) + c{i} = C{i}; + else + c{i} = parametrization.Curve.line(C{i}{:}); + end + end + + obj = parametrization.Ti(c{:}); end function label(varargin) @@ -190,11 +222,11 @@ ti.show(2,2); - grid.place_label(pc,str); - grid.place_label(pw,'w'); - grid.place_label(pe,'e'); - grid.place_label(ps,'s'); - grid.place_label(pn,'n'); + parametrization.place_label(pc,str); + parametrization.place_label(pw,'w'); + parametrization.place_label(pe,'e'); + parametrization.place_label(ps,'s'); + parametrization.place_label(pn,'n'); end end end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/curve_discretise.m --- a/+parametrization/old/curve_discretise.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/curve_discretise.m Tue Feb 07 16:09:02 2017 +0100 @@ -33,7 +33,7 @@ if do_plot fprintf('n:%d max: %f min: %f\n', n, max(d),min(d)); - p = grid.map_curve(g,t); + p = parametrization.map_curve(g,t); figure show(g,t,h); end @@ -44,11 +44,11 @@ t = linspace(0,1,n); t = equalize_d(g,t); d = D(g,t); - p = grid.map_curve(g,t); + p = parametrization.map_curve(g,t); end function d = D(g,t) - p = grid.map_curve(g,t); + p = parametrization.map_curve(g,t); d = zeros(1,length(t)-1); for i = 1:length(d) @@ -71,11 +71,11 @@ function show(g,t,hh) - p = grid.map_curve(g,t); + p = parametrization.map_curve(g,t); - h = grid.plot_curve(g); + h = parametrization.plot_curve(g); h.LineWidth = 2; axis equal hold on @@ -84,7 +84,7 @@ h.MarkerSize = 24; hold off - n = grid.curve_normals(g,t); + n = parametrization.curve_normals(g,t); hold on for i = 1:length(t) p0 = p(:,i);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/plot_shape.m --- a/+parametrization/old/plot_shape.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/plot_shape.m Tue Feb 07 16:09:02 2017 +0100 @@ -6,7 +6,7 @@ hs = {}; hold on for i = 1:length(s) - hs{end+1} = grid.plot_curve(s{i},n); + hs{end+1} = parametrization.plot_curve(s{i},n); end hold off end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/shape_discretise.m --- a/+parametrization/old/shape_discretise.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/shape_discretise.m Tue Feb 07 16:09:02 2017 +0100 @@ -2,7 +2,7 @@ function p = shape_discretise(s,h) p = []; for i = 1:length(s) - [~,pt] = grid.curve_discretise(s{i},h); + [~,pt] = parametrization.curve_discretise(s{i},h); p = [p, pt]; end end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/shape_linesegments.m --- a/+parametrization/old/shape_linesegments.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/shape_linesegments.m Tue Feb 07 16:09:02 2017 +0100 @@ -3,7 +3,7 @@ l = {}; for i = 1:length(s) - t = grid.curve_discretise(s{i},h); - l = [l, grid.curve_linesegments(s{i},t)]; + t = parametrization.curve_discretise(s{i},h); + l = [l, parametrization.curve_linesegments(s{i},t)]; end end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/triang_interp.m --- a/+parametrization/old/triang_interp.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/triang_interp.m Tue Feb 07 16:09:02 2017 +0100 @@ -13,9 +13,9 @@ o.A = g1(0); o.B = g2(0); o.C = g3(0); - o.Sa = grid.triang_interp.square_to_triangle_interp(g2,g3,g1); - o.Sb = grid.triang_interp.square_to_triangle_interp(g3,g1,g2); - o.Sc = grid.triang_interp.square_to_triangle_interp(g1,g2,g3); + o.Sa = parametrization.triang_interp.square_to_triangle_interp(g2,g3,g1); + o.Sb = parametrization.triang_interp.square_to_triangle_interp(g3,g1,g2); + o.Sc = parametrization.triang_interp.square_to_triangle_interp(g1,g2,g3); end @@ -30,17 +30,17 @@ ka = @(t)(o.g1(1-t)+o.g2(t))/2; - h = grid.plot_curve(ma); + h = parametrization.plot_curve(ma); h.Color = Color.blue; - h = grid.plot_curve(mb); + h = parametrization.plot_curve(mb); h.Color = Color.blue; - h = grid.plot_curve(mc); + h = parametrization.plot_curve(mc); h.Color = Color.blue; - h = grid.plot_curve(na); + h = parametrization.plot_curve(na); h.Color = Color.red; - h = grid.plot_curve(ka); + h = parametrization.plot_curve(ka); h.Color = Color.red; [a(1),a(2)] = ma(1/3); @@ -50,7 +50,7 @@ d = ka(1-1/3); - grid.label_pt(a,b,c,d); + parametrization.label_pt(a,b,c,d); % t = linspace(0,1,N); @@ -59,23 +59,23 @@ % sb = @(s)o.Sb(s,t(i)); % sc = @(s)o.Sc(s,t(i)); - % h = grid.plot_curve(sa); + % h = parametrization.plot_curve(sa); % h.Color = Color.blue; - % h = grid.plot_curve(sb); + % h = parametrization.plot_curve(sb); % h.Color = Color.blue; - % h = grid.plot_curve(sc); + % h = parametrization.plot_curve(sc); % h.Color = Color.blue; % end - h = grid.plot_curve(o.g1); + h = parametrization.plot_curve(o.g1); h.LineWidth = 2; h.Color = Color.red; - h = grid.plot_curve(o.g2); + h = parametrization.plot_curve(o.g2); h.LineWidth = 2; h.Color = Color.red; - h = grid.plot_curve(o.g3); + h = parametrization.plot_curve(o.g3); h.LineWidth = 2; h.Color = Color.red; @@ -90,8 +90,8 @@ % The collapsed side is mapped to the corner oposite to g1. % This is done such that for S(s,t), S(s,1) = g1(s) function S = square_to_triangle_interp(g1,g2,g3) - corner = grid.line_segment(g3(0),g3(0)); - S = grid.transfinite_interp(corner,g3,f(g1),f(g2)) + corner = parametrization.line_segment(g3(0),g3(0)); + S = parametrization.transfinite_interp(corner,g3,f(g1),f(g2)) % Function to flip a curve function h = f(g)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/triang_interp_pts.m --- a/+parametrization/old/triang_interp_pts.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/triang_interp_pts.m Tue Feb 07 16:09:02 2017 +0100 @@ -8,5 +8,5 @@ g2 = @(t)(p2 + t*(p3-p2)); g3 = @(t)(p3 + t*(p1-p3)); - S = grid.triang_interp(g1,g2,g3); + S = parametrization.triang_interp(g1,g2,g3); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +parametrization/old/triang_show.m --- a/+parametrization/old/triang_show.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+parametrization/old/triang_show.m Tue Feb 07 16:09:02 2017 +0100 @@ -4,8 +4,8 @@ ih = ishold(); hold on - h_grid = grid.triang_plot_interp(S,n); - h_bord = grid.triang_plot_interp(S,2); + h_grid = parametrization.triang_plot_interp(S,n); + h_bord = parametrization.triang_plot_interp(S,2); set(h_grid,'Color',[0 0.4470 0.7410]); set(h_bord,'Color',[0.8500 0.3250 0.0980]);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d1_upwind_3.m --- a/+sbp/+implementations/d1_upwind_3.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d1_upwind_3.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,12 +1,12 @@ function [H, HI, Dp, Dm, e_1, e_m] = d1_upwind_3(m,h) - + if(m<6) error('Operator requires at least 6 grid points'); end Hv = ones(m,1); Hv(1:3) = [3/8; 7/6; 23/24]; - Hv(m-2:m) = rot90(Hv(1:3),2); + Hv(m-2:m) = rot90(Hv(1:3), 2); Hv = Hv*h; H = spdiag(Hv,0); HI = spdiag(1./Hv,0);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d1_upwind_4.m --- a/+sbp/+implementations/d1_upwind_4.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d1_upwind_4.m Tue Feb 07 16:09:02 2017 +0100 @@ -23,7 +23,7 @@ ]; Qp(1:4,1:4)=Q_U; - Qp(m-3:m,m-3:m)=rot90(Q_U,2)'; %%% This is different from standard SBP + Qp(m-3:m,m-3:m)=rot90(Q_U, 2)'; %%% This is different from standard SBP Qm=-Qp';
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d2_2.m --- a/+sbp/+implementations/d2_2.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d2_2.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ function [H, HI, D1, D2, e_1, e_m, M, Q, S_1, S_m] = d2_2(m,h) - + BP = 1; if(m<2*BP) error(['Operator requires at least ' num2str(2*BP) ' grid points']); @@ -11,33 +11,40 @@ H=(speye(m,m));H(1,1)=0.5;H(m,m)=0.5; H=h*H; HI=inv(H); - + diags = -1:1; stencil = [-1/2 0 1/2]; D1 = stripeMatrix(stencil, diags, m); - - D1(1,1)=-1;D1(1,2)=1;D1(m,m-1)=-1;D1(m,m)=1; - D1(m,m-1)=-1;D1(m,m)=1; - D1=D1/h; + + D1(1,1) = -1; + D1(1,2) = 1; - Q=H*D1 + 1/2*(e_1*e_1') - 1/2*(e_m*e_m'); + D1(m,m-1) = -1; + D1(m,m) = 1; + + D1 = D1/h; + + Q = H*D1 + 1/2*(e_1*e_1') - 1/2*(e_m*e_m'); diags = -1:1; stencil = [1 -2 1]; D2 = stripeMatrix(stencil, diags, m); - - D2(1,1)=1;D2(1,2)=-2;D2(1,3)=1; - D2(m,m-2)=1;D2(m,m-1)=-2;D2(m,m)=1; - D2=D2/h^2; - S_U=[-3/2, 2, -1/2]/h; - S_1=sparse(1,m); - S_1(1:3)=S_U; - S_m=sparse(1,m); - S_m(m-2:m)=fliplr(-S_U); + D2(1,1) = 1; + D2(1,2) = -2; + D2(1,3) = 1; + D2(m,m-2) = 1; + D2(m,m-1) = -2; + D2(m,m) = 1; + D2 = D2/h^2; + S_U = [-3/2, 2, -1/2]/h; + S_1 = sparse(1,m); + S_1(1:3) = S_U; + S_m = sparse(1,m); + S_m(m-2:m) = fliplr(-S_U); - M=-H*D2-e_1*S_1+e_m*S_m; + M = -H*D2-e_1*S_1+e_m*S_m; S_1 = S_1'; S_m = S_m'; end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d2_4.m --- a/+sbp/+implementations/d2_4.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d2_4.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ function [H, HI, D1, D2, e_1, e_m, M, Q, S_1, S_m] = d2_4(m,h) - + BP = 4; if(m<2*BP) error(['Operator requires at least ' num2str(2*BP) ' grid points']); @@ -20,7 +20,12 @@ Q=spdiags([e -8*e 0*e 8*e -e], -2:2, m, m)/12; %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); - Q_U = [0 0.59e2 / 0.96e2 -0.1e1 / 0.12e2 -0.1e1 / 0.32e2; -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2; 0.1e1 / 0.32e2 0 -0.59e2 / 0.96e2 0;]; + Q_U = [ + 0 0.59e2/0.96e2 -0.1e1/0.12e2 -0.1e1/0.32e2; + -0.59e2/0.96e2 0 0.59e2/0.96e2 0; + 0.1e1/0.12e2 -0.59e2/0.96e2 0 0.59e2/0.96e2; + 0.1e1/0.32e2 0 -0.59e2/0.96e2 0; + ]; Q(1:4,1:4)=Q_U; Q(m-3:m,m-3:m)=rot90( -Q_U(1:4,1:4) ,2 ); @@ -30,13 +35,18 @@ %M=-(-1/12*diag(ones(m-2,1),2)+16/12*diag(ones(m-1,1),1)+16/12*diag(ones(m-1,1),-1)-1/12*diag(ones(m-2,1),-2)-30/12*diag(ones(m,1),0)); - M_U=[0.9e1 / 0.8e1 -0.59e2 / 0.48e2 0.1e1 / 0.12e2 0.1e1 / 0.48e2; -0.59e2 / 0.48e2 0.59e2 / 0.24e2 -0.59e2 / 0.48e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.48e2 0.55e2 / 0.24e2 -0.59e2 / 0.48e2; 0.1e1 / 0.48e2 0 -0.59e2 / 0.48e2 0.59e2 / 0.24e2;]; + M_U=[ + 0.9e1/0.8e1 -0.59e2/0.48e2 0.1e1/0.12e2 0.1e1/0.48e2; + -0.59e2/0.48e2 0.59e2/0.24e2 -0.59e2/0.48e2 0; + 0.1e1/0.12e2 -0.59e2/0.48e2 0.55e2/0.24e2 -0.59e2/0.48e2; + 0.1e1/0.48e2 0 -0.59e2/0.48e2 0.59e2/0.24e2; + ]; M(1:4,1:4)=M_U; M(m-3:m,m-3:m)=rot90( M_U ,2 ); M=M/h; - S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; + S_U=[-0.11e2/0.6e1 3 -0.3e1/0.2e1 0.1e1/0.3e1;]/h; S_1=sparse(sparse(1,m)); S_1(1:4)=S_U; S_m=sparse(sparse(1,m));
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d2_variable_4.m --- a/+sbp/+implementations/d2_variable_4.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d2_variable_4.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,6 +1,6 @@ function [H, HI, D1, D2, e_l, e_r, d1_l, d1_r] = d2_variable_4(m,h) - BP = 4; + BP = 6; if(m<2*BP) error(['Operator requires at least ' num2str(2*BP) ' grid points']); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_compatible_6.m --- a/+sbp/+implementations/d4_compatible_6.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/+implementations/d4_compatible_6.m Tue Feb 07 16:09:02 2017 +0100 @@ -25,7 +25,7 @@ % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm, % vilket ?r n?dv?ndigt f?r stabilitet - + BP = 8; if(m<2*BP) error(['Operator requires at least ' num2str(2*BP) ' grid points']);
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_compatible_halfvariable_2.m --- a/+sbp/+implementations/d4_compatible_halfvariable_2.m Tue Feb 07 15:47:51 2017 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,179 +0,0 @@ -% Returns D2 as a function handle -function [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1,... - S2_m, S3_1, S3_m, S_1, S_m] = d4_compatible_halfvariable_2(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 4:de ordn. SBP Finita differens %%% - %%% operatorer framtagna av Ken Mattsson %%% - %%% %%% - %%% 6 randpunkter, diagonal norm %%% - %%% %%% - %%% Datum: 2013-11-11 %%% - %%% %%% - %%% %%% - %%% H (Normen) %%% - %%% D1 (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%% D3 (approx tredje derivatan) %%% - %%% D2 (approx fj?rde derivatan) %%% - %%% %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - % M?ste ange antal punkter (m) och stegl?ngd (h) - % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r - % vi har 3de och 4de derivator i v?r PDE - % I annat fall anv?nd de "traditionella" som har noggrannare - % randsplutningar f?r D1 och D2 - - % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm, - % vilket ?r n?dv?ndigt f?r stabilitet - - BP = 4; - if(m<2*BP) - error(['Operator requires at least ' num2str(2*BP) ' grid points']); - end - - H=speye(m,m);H(1,1)=1/2;H(m,m)=1/2; - - - H=H*h; - HI=inv(H); - - - % First derivative SBP operator, 1st order accurate at first 6 boundary points - - q1=1/2; -% Q=q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); - stencil = [-q1,0,q1]; - d = (length(stencil)-1)/2; - diags = -d:d; - Q = stripeMatrix(stencil, diags, m); - - %Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); - - - e_1=sparse(m,1);e_1(1)=1; - e_m=sparse(m,1);e_m(m)=1; - - - D1=HI*(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ; - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - % Second derivative, 1st order accurate at first boundary points - - % below for constant coefficients - % m1=-1;m0=2; - % M=m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0);M(1,1)=1;M(m,m)=1; - % M=M/h; - %D2=HI*(-M-e_1*S_1+e_m*S_m); - - % Below for variable coefficients - % Require a vector c with the koeffients - - S_U=[-3/2 2 -1/2]/h; - S_1=sparse(1,m); - S_1(1:3)=S_U; - S_m=sparse(1,m); - S_m(m-2:m)=fliplr(-S_U); - - S_1 = S_1'; - S_m = S_m'; - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - scheme_width = 3; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - Mm1 = -c(r-1)/2 - c(r)/2; - M0 = c(r-1)/2 + c(r) + c(r+1)/2; - Mp1 = -c(r)/2 - c(r+1)/2; - - M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m); - - - M(1:2,1:2)=[c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;]; - M(m-1:m,m-1:m)=[c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;]; - M=M/h; - - D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); - end - D2 = @D2_fun; - - - - - - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - % Third derivative, 1st order accurate at first 6 boundary points - - q2=1/2;q1=-1; -% Q3=q2*(diag(ones(m-2,1),2)-diag(ones(m-2,1),-2))+q1*(diag(ones(m-1,1),1)-diag(ones(m-1,1),-1)); - stencil = [-q2,-q1,0,q1,q2]; - d = (length(stencil)-1)/2; - diags = -d:d; - Q3 = stripeMatrix(stencil, diags, m); - - %QQ3=(-1/8*diag(ones(m-3,1),3) + 1*diag(ones(m-2,1),2) - 13/8*diag(ones(m-1,1),1) +13/8*diag(ones(m-1,1),-1) -1*diag(ones(m-2,1),-2) + 1/8*diag(ones(m-3,1),-3)); - - - Q3_U = [0 -0.13e2 / 0.16e2 0.7e1 / 0.8e1 -0.1e1 / 0.16e2; 0.13e2 / 0.16e2 0 -0.23e2 / 0.16e2 0.5e1 / 0.8e1; -0.7e1 / 0.8e1 0.23e2 / 0.16e2 0 -0.17e2 / 0.16e2; 0.1e1 / 0.16e2 -0.5e1 / 0.8e1 0.17e2 / 0.16e2 0;]; - Q3(1:4,1:4)=Q3_U; - Q3(m-3:m,m-3:m)=rot90( -Q3_U ,2 ); - Q3=Q3/h^2; - - - - S2_U=[1 -2 1;]/h^2; - S2_1=sparse(1,m); - S2_1(1:3)=S2_U; - S2_m=sparse(1,m); - S2_m(m-2:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - - - D3=HI*(Q3 - e_1*S2_1' + e_m*S2_m' +1/2*(S_1*S_1') -1/2*(S_m*S_m') ) ; - - % Fourth derivative, 0th order accurate at first 6 boundary points (still - % yield 4th order convergence if stable: for example u_tt=-u_xxxx - - m2=1;m1=-4;m0=6; -% M4=m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - stencil = [m2,m1,m0,m1,m2]; - d = (length(stencil)-1)/2; - diags = -d:d; - M4 = stripeMatrix(stencil, diags, m); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.13e2 / 0.10e2 -0.12e2 / 0.5e1 0.9e1 / 0.10e2 0.1e1 / 0.5e1; -0.12e2 / 0.5e1 0.26e2 / 0.5e1 -0.16e2 / 0.5e1 0.2e1 / 0.5e1; 0.9e1 / 0.10e2 -0.16e2 / 0.5e1 0.47e2 / 0.10e2 -0.17e2 / 0.5e1; 0.1e1 / 0.5e1 0.2e1 / 0.5e1 -0.17e2 / 0.5e1 0.29e2 / 0.5e1;]; - - - M4(1:4,1:4)=M4_U; - - M4(m-3:m,m-3:m)=rot90( M4_U ,2 ); - M4=M4/h^3; - - S3_U=[-1 3 -3 1;]/h^3; - S3_1=sparse(1,m); - S3_1(1:4)=S3_U; - S3_m=sparse(1,m); - S3_m(m-3:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); -end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_compatible_halfvariable_4.m --- a/+sbp/+implementations/d4_compatible_halfvariable_4.m Tue Feb 07 15:47:51 2017 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,167 +0,0 @@ -function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1,... - S3_m, S_1, S_m] = d4_compatible_halfvariable_4(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 4:de ordn. SBP Finita differens %%% - %%% %%% - %%% H (Normen) %%% - %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - %m=20; %problemstorlek - %h=1/(m-1); - %h=1; - - BP = 6; - if(m<2*BP) - error(['Operator requires at least ' num2str(2*BP) ' grid points']); - end - -% c=ones(m,1); - - - H=speye(m,m); - H(1:4,1:4)=diag([17/48 59/48 43/48 49/48]); - H(m-3:m,m-3:m)=rot90(diag([17/48 59/48 43/48 49/48]),2); - H=H*h; - HI=inv(H); - HI = sparse(HI); - - - -% Q=(-1/12*diag(ones(m-2,1),2)+8/12*diag(ones(m-1,1),1)-8/12*diag(ones(m-1,1),-1)+1/12*diag(ones(m-2,1),-2)); - e=ones(m,1); -% Q=spdiags([e -8*e 0*e 8*e -e], -2:2, m, m)/12; -% Q_U = [0 0.59e2 / 0.96e2 -0.1e1 / 0.12e2 -0.1e1 / 0.32e2; -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.96e2 0 0.59e2 / 0.96e2; 0.1e1 / 0.32e2 0 -0.59e2 / 0.96e2 0;]; -% Q(1:4,1:4)=Q_U; -% Q(m-3:m,m-3:m)=rot90( -Q_U(1:4,1:4) ,2 ); - - e_1=sparse(m,1);e_1(1)=1; - e_m=sparse(m,1);e_m(m)=1; - -% D1=HI*(Q-1/2*(e_1*e_1')+1/2*(e_m*e_m')) ; - - M_U=[0.9e1 / 0.8e1 -0.59e2 / 0.48e2 0.1e1 / 0.12e2 0.1e1 / 0.48e2; -0.59e2 / 0.48e2 0.59e2 / 0.24e2 -0.59e2 / 0.48e2 0; 0.1e1 / 0.12e2 -0.59e2 / 0.48e2 0.55e2 / 0.24e2 -0.59e2 / 0.48e2; 0.1e1 / 0.48e2 0 -0.59e2 / 0.48e2 0.59e2 / 0.24e2;]; -% M=-(-1/12*diag(ones(m-2,1),2)+16/12*diag(ones(m-1,1),1)+16/12*diag(ones(m-1,1),-1)-1/12*diag(ones(m-2,1),-2)-30/12*diag(ones(m,1),0)); - M=-spdiags([-e 16*e -30*e 16*e -e], -2:2, m, m)/12; - - M(1:4,1:4)=M_U; - - M(m-3:m,m-3:m)=rot90( M_U ,2 ); - M=M/h; - - S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; - S_1=sparse(1,m); - S_1(1:4)=S_U; - S_m=sparse(1,m); - S_m(m-3:m)=fliplr(-S_U); - S_1 = S_1'; - S_m = S_m'; - - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - - scheme_width = 5; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - % ALTERNATIVES %%%%%%%%%%%%% - % for i=4:m-3 - % M(i,i-2:i+2)=[-c(i-1) / 0.6e1 + c(i-2) / 0.8e1 + c(i) / 0.8e1 -c(i-2) / 0.6e1 - c(i+1) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 c(i-2) / 0.24e2 + 0.5e1 / 0.6e1 * c(i-1) + 0.5e1 / 0.6e1 * c(i+1) + c(i+2) / 0.24e2 + 0.3e1 / 0.4e1 * c(i) -c(i-1) / 0.6e1 - c(i+2) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 -c(i+1) / 0.6e1 + c(i) / 0.8e1 + c(i+2) / 0.8e1;]; - % end - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % for i=4:m-3 - % M(i,i-2:i+2)= [ - % c(i-2) / 0.8e1 - c(i-1) / 0.6e1 + c(i) / 0.8e1 , - % -c(i-2) / 0.6e1 - c(i-1) / 0.2e1 - c(i) / 0.2e1 - c(i+1) / 0.6e1 , - % c(i-2) / 2.4e1 + c(i-1) / 1.2e0 + c(i) * 0.3/0.4 + c(i+1) / 1.2e0 + c(i+2) / 2.4e1 , - % -c(i-1) / 0.6e1 - c(i) / 0.2e1 - c(i+1) / 0.2e1 - c(i+2) / 0.6e1 , - % c(i) / 0.8e1 - c(i+1) / 0.6e1 + c(i+2) / 0.8e1 , - % ]; - % end - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; - Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; - M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; - Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; - Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; - % printSize(Mm2); - % scheme_radius - % m - M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m); - % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],(-2:2)+scheme_radius,M(r,:)); % This is slower - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % %% Somthing is wrong here!! - % Mm2 = c(r-2) / 0.8e1 - c(r-1) / 0.6e1 + c(r) / 0.8e1 ; - % Mm1 = -c(r-2) / 0.6e1 - c(r-1) / 0.2e1 - c(r) / 0.2e1 - c(r+1) / 0.6e1 ; - % M0 = c(r-2) / 2.4e1 + c(r-1) / 1.2e0 + c(r) * 0.3/0.4 + c(r+1) / 1.2e0 + c(r+2) / 2.4e1; - % Mp1 = -c(r-1) / 0.6e1 - c(r) / 0.2e1 - c(r+1) / 0.2e1 - c(r+2) / 0.6e1; - % Mp2 = c(r) / 0.8e1 - c(r+1) / 0.6e1 + c(r+2) / 0.8e1; - % % printSize(M_diag_ind); - % % Mdiags = [Mm2 Mm1 M0 Mp1 Mp2]; - % % printSize(Mdiags); - % M(M_diag_ind) = [Mm2 Mm1 M0 Mp1 Mp2]; % This is slightly faster - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - % Kan man skriva det som en multiplikation av en 3-dim matris? - %%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - - M(1:6,1:6)=[0.12e2 / 0.17e2 * c(1) + 0.59e2 / 0.192e3 * c(2) + 0.27010400129e11 / 0.345067064608e12 * c(3) + 0.69462376031e11 / 0.2070402387648e13 * c(4) -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -c(4) / 0.784e3 + c(3) / 0.784e3; -0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4) 0.3481e4 / 0.3264e4 * c(1) + 0.9258282831623875e16 / 0.7669235228057664e16 * c(3) + 0.236024329996203e15 / 0.1278205871342944e16 * c(4) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4); 0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3) -0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4) c(1) / 0.51e2 + 0.59e2 / 0.192e3 * c(2) + 0.13777050223300597e17 / 0.26218083221499456e17 * c(4) + 0.564461e6 / 0.13384296e8 * c(5) + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(3) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3); 0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4) -0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4) c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4) 0.3e1 / 0.1088e4 * c(1) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(3) + 0.1869103e7 / 0.2230716e7 * c(5) + c(6) / 0.24e2 + 0.1950062198436997e16 / 0.3834617614028832e16 * c(4) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1; 0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4) -0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4) -0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5) -0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5) 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(3) + 0.2224717261773437e16 / 0.2763180339520776e16 * c(4) + 0.5e1 / 0.6e1 * c(6) + c(7) / 0.24e2 + 0.280535e6 / 0.371786e6 * c(5) -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1; -c(4) / 0.784e3 + c(3) / 0.784e3 -0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4) -0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3) -0.368395e6 / 0.2230716e7 * c(5) + 0.752806667e9 / 0.539854092016e12 * c(3) + 0.1063649e7 / 0.8712336e7 * c(4) + c(6) / 0.8e1 -0.35039615e8 / 0.213452232e9 * c(4) - c(7) / 0.6e1 - 0.13091810925e11 / 0.13226425254392e14 * c(3) - 0.1118749e7 / 0.2230716e7 * c(5) - c(6) / 0.2e1 0.3290636e7 / 0.80044587e8 * c(4) + 0.5580181e7 / 0.6692148e7 * c(5) + 0.5e1 / 0.6e1 * c(7) + c(8) / 0.24e2 + 0.660204843e9 / 0.13226425254392e14 * c(3) + 0.3e1 / 0.4e1 * c(6);]; - - M(m-5:m,m-5:m)=[c(m-7) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-6) + 0.5580181e7 / 0.6692148e7 * c(m-4) + 0.4887707739997e13 / 0.119037827289528e15 * c(m-3) + 0.3e1 / 0.4e1 * c(m-5) + 0.660204843e9 / 0.13226425254392e14 * c(m-2) + 0.660204843e9 / 0.13226425254392e14 * c(m-1) -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3; -c(m-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(m-3) - c(m-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(m-4) - 0.13091810925e11 / 0.13226425254392e14 * c(m-2) - 0.13091810925e11 / 0.13226425254392e14 * c(m-1) c(m-6) / 0.24e2 + 0.5e1 / 0.6e1 * c(m-5) + 0.3896014498639e13 / 0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(m-2) + 0.280535e6 / 0.371786e6 * c(m-4) + 0.3360696339136261875e19 / 0.171218627618178537656e21 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1); -0.368395e6 / 0.2230716e7 * c(m-4) + c(m-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(m-3) + 0.752806667e9 / 0.539854092016e12 * c(m-2) + 0.752806667e9 / 0.539854092016e12 * c(m-1) -c(m-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(m-2) - 0.375177e6 / 0.743572e6 * c(m-4) - 0.13425842714e11 / 0.33740880751e11 * c(m-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(m-1) c(m-5) / 0.24e2 + 0.1869103e7 / 0.2230716e7 * c(m-4) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(m-2) + 0.3e1 / 0.1088e4 * c(m) + 0.31688435395e11 / 0.67481761502e11 * c(m-3) + 0.27769176016102795561e20 / 0.712828572124661666976e21 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1); -0.3391e4 / 0.6692148e7 * c(m-4) - 0.238797444493e12 / 0.119037827289528e15 * c(m-3) + 0.33235054191e11 / 0.26452850508784e14 * c(m-2) + 0.33235054191e11 / 0.26452850508784e14 * c(m-1) -0.365281640980e12 / 0.1653303156799e13 * c(m-3) + 0.564461e6 / 0.4461432e7 * c(m-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(m-1) -0.125059e6 / 0.743572e6 * c(m-4) + c(m) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(m-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(m-1) 0.564461e6 / 0.13384296e8 * c(m-4) + 0.470299699916357e15 / 0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21 / 0.1624586048098066124736e22 * c(m-1) + c(m) / 0.51e2 + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(m-2) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2); -0.8673e4 / 0.2904112e7 * c(m-2) - 0.8673e4 / 0.2904112e7 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.1328188692663e13 / 0.37594290333616e14 * c(m-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.260297319232891e15 / 0.2556411742685888e16 * c(m-2) - 0.59e2 / 0.1088e4 * c(m) - 0.106641839640553e15 / 0.1278205871342944e16 * c(m-1) + 0.26019e5 / 0.726028e6 * c(m-3) -0.59e2 / 0.408e3 * c(m) - 0.29294615794607e14 / 0.29725717938208e14 * c(m-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(m-1) - 0.8673e4 / 0.363014e6 * c(m-3) 0.9258282831623875e16 / 0.7669235228057664e16 * c(m-2) + 0.3481e4 / 0.3264e4 * c(m) + 0.228389721191751e15 / 0.1278205871342944e16 * c(m-1) + 0.8673e4 / 0.1452056e7 * c(m-3) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1); -c(m-3) / 0.392e3 + c(m-2) / 0.784e3 + c(m-1) / 0.784e3 c(m-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(m-2) - 0.256702175e9 / 0.10149031312e11 * c(m-1) -0.1244724001e10 / 0.21126554976e11 * c(m-2) + 0.3e1 / 0.68e2 * c(m) + 0.752806667e9 / 0.21126554976e11 * c(m-1) -0.59e2 / 0.3136e4 * c(m-3) - 0.13249937023e11 / 0.48148892736e11 * c(m-1) + 0.2e1 / 0.17e2 * c(m) + 0.2083938599e10 / 0.8024815456e10 * c(m-2) -0.6025413881e10 / 0.21126554976e11 * c(m-2) - 0.59e2 / 0.68e2 * c(m) - 0.537416663e9 / 0.7042184992e10 * c(m-1) 0.3e1 / 0.3136e4 * c(m-3) + 0.27010400129e11 / 0.345067064608e12 * c(m-2) + 0.234566387291e12 / 0.690134129216e12 * c(m-1) + 0.12e2 / 0.17e2 * c(m);]; - - M=M/h; - D2=HI*(-M-c(1)*e_1*S_1'+c(m)*e_m*S_m'); - end - D2 = @D2_fun; - - - S2_U=[2 -5 4 -1;]/h^2; - S2_1=sparse(1,m); - S2_1(1:4)=S2_U; - S2_m=sparse(1,m); - S2_m(m-3:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - m3=-1/6;m2=2;m1=-13/2;m0=28/3; -% M4=m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - stencil = [m3,m2,m1,m0,m1,m2,m3]; - d = (length(stencil)-1)/2; - diags = -d:d; - M4 = stripeMatrix(stencil, diags, m); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.5762947e7 / 0.2316384e7 -0.6374287e7 / 0.1158192e7 0.573947e6 / 0.165456e6 -0.124637e6 / 0.289548e6 0.67979e5 / 0.2316384e7 -0.60257e5 / 0.1158192e7; -0.6374287e7 / 0.1158192e7 0.30392389e8 / 0.2316384e7 -0.2735053e7 / 0.289548e6 0.273109e6 / 0.165456e6 0.83767e5 / 0.1158192e7 0.245549e6 / 0.2316384e7; 0.573947e6 / 0.165456e6 -0.2735053e7 / 0.289548e6 0.5266855e7 / 0.579096e6 -0.1099715e7 / 0.289548e6 0.869293e6 / 0.1158192e7 -0.10195e5 / 0.144774e6; -0.124637e6 / 0.289548e6 0.273109e6 / 0.165456e6 -0.1099715e7 / 0.289548e6 0.3259225e7 / 0.579096e6 -0.324229e6 / 0.72387e5 0.1847891e7 / 0.1158192e7; 0.67979e5 / 0.2316384e7 0.83767e5 / 0.1158192e7 0.869293e6 / 0.1158192e7 -0.324229e6 / 0.72387e5 0.2626501e7 / 0.330912e6 -0.7115491e7 / 0.1158192e7; -0.60257e5 / 0.1158192e7 0.245549e6 / 0.2316384e7 -0.10195e5 / 0.144774e6 0.1847891e7 / 0.1158192e7 -0.7115491e7 / 0.1158192e7 0.21383077e8 / 0.2316384e7;]; - - M4(1:6,1:6)=M4_U; - - M4(m-5:m,m-5:m)=rot90( M4_U ,2 ); - M4=M4/h^3; - - S3_U=[-1 3 -3 1;]/h^3; - S3_1=sparse(1,m); - S3_1(1:4)=S3_U; - S3_m=sparse(1,m); - S3_m(m-3:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); - - - - - -end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_compatible_halfvariable_6.m --- a/+sbp/+implementations/d4_compatible_halfvariable_6.m Tue Feb 07 15:47:51 2017 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,170 +0,0 @@ -function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1,... - S3_m, S_1, S_m] = d4_compatible_halfvariable_6(m,h) - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - %%% 6:te ordn. SBP Finita differens %%% - %%% operatorer med diagonal norm %%% - %%% Extension to variable koeff %%% - %%% %%% - %%% H (Normen) %%% - %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% - %%% D2 (approx andra derivatan) %%% - %%% D2=HI*(R+C*D*S %%% - %%% %%% - %%% R=-D1'*H*C*D1-RR %%% - %%% %%% - %%% RR ?r dissipation) %%% - %%% Dissipationen uppbyggd av D4: %%% - %%% DI=D4*B*H*D4 %%% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - %m=10; %problemstorlek - %h=1/(m-1); - - % Variable koefficicients are stored in vector: c, size m, - % with the unknown stored as c(1), c(2), ..., c_m - % x=1:h:m*h;x=x'; - % c=x.^0; - - BP = 8; - if(m<2*BP) - error(['Operator requires at least ' num2str(2*BP) ' grid points']); - end - - - H=speye(m,m); - H(1:6,1:6)=diag([13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, ... - 43801/43200]); - H(m-5:m,m-5:m)=rot90(diag([13649/43200,12013/8640, ... - 2711/4320,5359/4320,7877/8640,43801/43200]),2); - - -% x1=0.70127127127127; - - -% D1=(1/60*diag(ones(m-3,1),3)-9/60*diag(ones(m-2,1),2)+45/60*diag(ones(m-1,1),1)-45/60*diag(ones(m-1,1),-1)+9/60*diag(ones(m-2,1),-2)-1/60*diag(ones(m-3,1),-3)); -% -% -% -% D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ... -% 715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ... -% x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ... -% -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ... -% -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ... -% -34560/12013*x1+240569/120130, 0, 0, 0; ... -% 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, ... -% 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, ... -% 25920/2711*x1-182261/27110, 0, 0, 0; ... -% -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, ... -% -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, ... -% -17280/5359*x1+710473/321540, 72/5359, 0, 0; ... -% 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, ... -% 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, ... -% 0, 8640/7877*x1, -1296/7877, 144/7877, 0; ... -% -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602,... -% -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, ... -% -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801]; -% D1(m-5:m,m-8:m)=rot90( -D1(1:6,1:9),2); -% D1=D1/h; - - e_1=sparse(m,1);e_1(1)=1; - e_m=sparse(m,1);e_m(m)=1; - - S_U=[-25/12, 4, -3, 4/3, -1/4]/h; - S_1=sparse(1,m); - S_1(1:5)=S_U; - S_m=sparse(1,m); - S_m(m-4:m)=fliplr(-S_U); - S_1 = S_1'; - S_m = S_m'; - - - - %DS=sparse(m,m); - %DS(1,1:5)=-[-25/12, 4, -3, 4/3, -1/4]; - %DS(m,m-4:m)=fliplr(-[-25/12, 4, -3, 4/3, -1/4]); - %DS=diag(c)*DS/h; - - - H=h*H; - HI=inv(H); - - - M=sparse(m,m); - e_1 = sparse(e_1); - e_m = sparse(e_m); - S_1 = sparse(S_1); - S_m = sparse(S_m); - - scheme_width = 7; - scheme_radius = (scheme_width-1)/2; - r = (1+scheme_radius):(m-scheme_radius); - - function D2 = D2_fun(c) - - Mm3 = c(r-2) / 0.40e2 + c(r-1) / 0.40e2 - 0.11e2 / 0.360e3 * c(r-3) - 0.11e2 / 0.360e3 * c(r); - Mm2 = c(r-3) / 0.20e2 - 0.3e1 / 0.10e2 * c(r-1) + c(r+1) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r-2); - Mm1 = -c(r-3) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-2) - 0.3e1 / 0.10e2 * c(r+1) - c(r+2) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r-1); - M0 = c(r-3) / 0.180e3 + c(r-2) / 0.8e1 + 0.19e2 / 0.20e2 * c(r-1) + 0.19e2 / 0.20e2 * c(r+1) + c(r+2) / 0.8e1 + c(r+3) / 0.180e3 + 0.101e3 / 0.180e3 * c(r); - Mp1 = -c(r-2) / 0.40e2 - 0.3e1 / 0.10e2 * c(r-1) - 0.3e1 / 0.10e2 * c(r+2) - c(r+3) / 0.40e2 - 0.17e2 / 0.40e2 * c(r) - 0.17e2 / 0.40e2 * c(r+1); - Mp2 = c(r-1) / 0.20e2 - 0.3e1 / 0.10e2 * c(r+1) + c(r+3) / 0.20e2 + 0.7e1 / 0.40e2 * c(r) + 0.7e1 / 0.40e2 * c(r+2); - Mp3 = c(r+1) / 0.40e2 + c(r+2) / 0.40e2 - 0.11e2 / 0.360e3 * c(r) - 0.11e2 / 0.360e3 * c(r+3); - - M(r,:) = spdiags([Mm3 Mm2 Mm1 M0 Mp1 Mp2 Mp3],0:2*scheme_radius,length(r),m); - - - M(1:9,1:9)=[0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); 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0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9);]; - - M(m-8:m,m-8:m)=[0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6);]; - - M(5,10)=M(10,5); - M(m-4,m-9)=M(m-9,m-4); - - M=M/h; - - D2=HI*(-M-diag(c)*e_1*S_1'+diag(c)*e_m*S_m'); - end - D2 = @D2_fun; - - S2_U=[0.35e2 / 0.12e2 -0.26e2 / 0.3e1 0.19e2 / 0.2e1 -0.14e2 / 0.3e1 0.11e2 / 0.12e2;]/h^2; - S2_1=sparse(1,m); - S2_1(1:5)=S2_U; - S2_m=sparse(1,m); - S2_m(m-4:m)=fliplr(S2_U); - S2_1 = S2_1'; - S2_m = S2_m'; - - - - - - % Fourth derivative, 1th order accurate at first 8 boundary points (still - % yield 5th order convergence if stable: for example u_tt=-u_xxxx - - m4=7/240;m3=-2/5;m2=169/60;m1=-122/15;m0=91/8; -% M4=m4*(diag(ones(m-4,1),4)+diag(ones(m-4,1),-4))+m3*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3))+m2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2))+m1*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1))+m0*diag(ones(m,1),0); - stencil = [m4,m3,m2,m1,m0,m1,m2,m3,m4]; - d = (length(stencil)-1)/2; - diags = -d:d; - M4 = stripeMatrix(stencil, diags, m); - - %M4=(-1/6*(diag(ones(m-3,1),3)+diag(ones(m-3,1),-3) ) + 2*(diag(ones(m-2,1),2)+diag(ones(m-2,1),-2)) -13/2*(diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)) + 28/3*diag(ones(m,1),0)); - - M4_U=[0.1394226315049e13 / 0.367201486080e12 -0.1137054563243e13 / 0.114750464400e12 0.16614189027367e14 / 0.1836007430400e13 -0.1104821700277e13 / 0.306001238400e12 0.1355771086763e13 / 0.1836007430400e13 -0.27818686453e11 / 0.459001857600e12 -0.40671054239e11 / 0.1836007430400e13 0.5442887371e10 / 0.306001238400e12; -0.1137054563243e13 / 0.114750464400e12 0.70616795535409e14 / 0.2570410402560e13 -0.173266854731041e15 / 0.6426026006400e13 0.28938615291031e14 / 0.2570410402560e13 -0.146167361863e12 / 0.71400288960e11 0.2793470836571e13 / 0.12852052012800e14 0.6219558097e10 / 0.428401733760e12 -0.7313844559e10 / 0.166909766400e12; 0.16614189027367e14 / 0.1836007430400e13 -0.173266854731041e15 / 0.6426026006400e13 0.378613061504779e15 / 0.12852052012800e14 -0.9117069604217e13 / 0.642602600640e12 0.632177582849e12 / 0.233673672960e12 -0.1057776382577e13 / 0.6426026006400e13 0.443019868399e12 / 0.4284017337600e13 -0.3707981e7 / 0.2318191200e10; -0.1104821700277e13 / 0.306001238400e12 0.28938615291031e14 / 0.2570410402560e13 -0.9117069604217e13 / 0.642602600640e12 0.5029150721885e13 / 0.514082080512e12 -0.5209119714341e13 / 0.1285205201280e13 0.12235427457469e14 / 0.12852052012800e14 -0.13731270505e11 / 0.64260260064e11 0.2933596129e10 / 0.40800165120e11; 0.1355771086763e13 / 0.1836007430400e13 -0.146167361863e12 / 0.71400288960e11 0.632177582849e12 / 0.233673672960e12 -0.5209119714341e13 / 0.1285205201280e13 0.14871726798559e14 / 0.2570410402560e13 -0.7504337615347e13 / 0.1606506501600e13 0.310830296467e12 / 0.171360693504e12 -0.55284274391e11 / 0.183600743040e12; -0.27818686453e11 / 0.459001857600e12 0.2793470836571e13 / 0.12852052012800e14 -0.1057776382577e13 / 0.6426026006400e13 0.12235427457469e14 / 0.12852052012800e14 -0.7504337615347e13 / 0.1606506501600e13 0.106318657014853e15 / 0.12852052012800e14 -0.14432772918527e14 / 0.2142008668800e13 0.58102695589e11 / 0.22666758400e11; -0.40671054239e11 / 0.1836007430400e13 0.6219558097e10 / 0.428401733760e12 0.443019868399e12 / 0.4284017337600e13 -0.13731270505e11 / 0.64260260064e11 0.310830296467e12 / 0.171360693504e12 -0.14432772918527e14 / 0.2142008668800e13 0.27102479467823e14 / 0.2570410402560e13 -0.1216032192203e13 / 0.153000619200e12; 0.5442887371e10 / 0.306001238400e12 -0.7313844559e10 / 0.166909766400e12 -0.3707981e7 / 0.2318191200e10 0.2933596129e10 / 0.40800165120e11 -0.55284274391e11 / 0.183600743040e12 0.58102695589e11 / 0.22666758400e11 -0.1216032192203e13 / 0.153000619200e12 0.20799922829107e14 / 0.1836007430400e13;]; - - M4(1:8,1:8)=M4_U; - - M4(m-7:m,m-7:m)=rot90( M4_U ,2 ); - M4=M4/h^3; - - S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; - S3_1=sparse(1,m); - S3_1(1:5)=S3_U; - S3_m=sparse(1,m); - S3_m(m-4:m)=fliplr(-S3_U); - S3_1 = S3_1'; - S3_m = S3_m'; - - D4=HI*(M4-e_1*S3_1'+e_m*S3_m' + S_1*S2_1'-S_m*S2_m'); - -end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_4_min_boundary_points.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_4_min_boundary_points.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,81 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_4_min_boundary_points(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 4:de ordn. SBP Finita differens %%% + %%% operatorer framtagna av Mark Carpenter %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %H?r med endast 4 randpunkter + + + BP = 4; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + + % Norm + Hv = ones(m,1); + Hv(1:4) = [17/48 59/48 43/48 49/48]; + Hv(m-3:m) = rot90(Hv(1:4),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:4) = 1/h*[-11/6 3 -3/2 1/3]; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:4) = 1/h^2*[2 -5 4 -1]; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:4) = 1/h^3*[-1 3 -3 1]; + d3_r = -rot90(d3_l, 2); + + + % First derivative + stencil = [1/12 -2/3 0 2/3 -1/12]; + diags = [-1 0 1]; + + Q_U = [ + 0 0.59e2/0.96e2 -0.1e1/0.12e2 -0.1e1/0.32e2; + -0.59e2/0.96e2 0 0.59e2/0.96e2 0; + 0.1e1/0.12e2 -0.59e2/0.96e2 0 0.59e2/0.96e2; + 0.1e1/0.32e2 0 -0.59e2/0.96e2 0; + ]; + + Q = stripeMatrix(stencil, diags, m); + Q(1:4,1:4)=Q_U; + Q(m-3:m,m-3:m) = -rot90(Q_U, 2); + + D1 = HI*(Q - 1/2*e_l*e_l' + 1/2*e_r*e_r'); + + % Fourth derivative + stencil = [-1/6, 2, -13/2, 28/3, -13/2, 2, -1/6]; + diags = -3:3; + M4 = stripeMatrix(stencil, diags, m); + + M4_U=[ + 0.8e1/0.3e1 -0.37e2/0.6e1 0.13e2/0.3e1 -0.5e1/0.6e1; + -0.37e2/0.6e1 0.47e2/0.3e1 -13 0.11e2/0.3e1; + 0.13e2/0.3e1 -13 0.44e2/0.3e1 -0.47e2/0.6e1; + -0.5e1/0.6e1 0.11e2/0.3e1 -0.47e2/0.6e1 0.29e2/0.3e1; + ]; + + + M4(1:4,1:4) = M4_U; + M4(m-3:m,m-3:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_6_2.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_6_2.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,76 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_6_2(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 6:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% Extension to variable koeff %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % H?r med 6 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator + % Denna ?r noggrannare, och har 2a ordningens randdslutning och b?r ge 6te + % ordningens konvergens. Hade dock ingen fri parameter att optimera + + BP = 6; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:6) = [0.181e3/0.576e3, 0.1343e4/0.960e3, 0.293e3/0.480e3, 0.1811e4/0.1440e4, 0.289e3/0.320e3, 0.65e2/0.64e2]; + Hv(m-5:m) = rot90(Hv(1:6),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:6) = [-0.137e3/0.60e2 5 -5 0.10e2/0.3e1 -0.5e1/0.4e1 0.1e1/0.5e1;]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:6) = [0.15e2/0.4e1 -0.77e2/0.6e1 0.107e3/0.6e1 -13 0.61e2/0.12e2 -0.5e1/0.6e1;]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:6) = [-0.17e2/0.4e1 0.71e2/0.4e1 -0.59e2/0.2e1 0.49e2/0.2e1 -0.41e2/0.4e1 0.7e1/0.4e1;]/h^3; + d3_r = -rot90(d3_l, 2); + + + % Fourth derivative, 1th order accurate at first 8 boundary points (still + % yield 5th order convergence if stable: for example u_tt = -u_xxxx + stencil = [7/240, -2/5, 169/60, -122/15, 91/8, -122/15, 169/60, -2/5, 7/240]; + diags = -4:4; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.1009e4/0.192e3 -0.7657e4/0.480e3 0.9307e4/0.480e3 -0.509e3/0.40e2 0.4621e4/0.960e3 -0.25e2/0.32e2; + -0.7657e4/0.480e3 0.49513e5/0.960e3 -0.4007e4/0.60e2 0.21799e5/0.480e3 -0.8171e4/0.480e3 0.2657e4/0.960e3; + 0.9307e4/0.480e3 -0.4007e4/0.60e2 0.1399e4/0.15e2 -0.2721e4/0.40e2 0.12703e5/0.480e3 -0.521e3/0.120e3; + -0.509e3/0.40e2 0.21799e5/0.480e3 -0.2721e4/0.40e2 0.3349e4/0.60e2 -0.389e3/0.15e2 0.559e3/0.96e2; + 0.4621e4/0.960e3 -0.8171e4/0.480e3 0.12703e5/0.480e3 -0.389e3/0.15e2 0.17857e5/0.960e3 -0.1499e4/0.160e3; + -0.25e2/0.32e2 0.2657e4/0.960e3 -0.521e3/0.120e3 0.559e3/0.96e2 -0.1499e4/0.160e3 0.2225e4/0.192e3; + ]; + + + M4(1:6,1:6) = M4_U; + M4(m-5:m,m-5:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_6_3.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_6_3.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,70 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_6_3(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 6:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% Extension to variable koeff %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % H?r med 7 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator + % Denna ?r noggrannare, och har 2a ordningens randdslutning och b?r ge 6te + % ordningens konvergens. Hade 2 fria parametrar att optimera + + % Norm + Hv = ones(m,1); + Hv(1:7) = [0.414837907e9/0.1191965760e10, 0.475278367e9/0.397321920e9, 0.13872751e8/0.12416310e8, 0.346739027e9/0.595982880e9, 0.560227469e9/0.397321920e9, 0.322971631e9/0.397321920e9, 0.616122491e9/0.595982880e9]; + Hv(m-6:m) = rot90(Hv(1:7),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:6) = [-0.137e3/0.60e2 5 -5 0.10e2/0.3e1 -0.5e1/0.4e1 0.1e1/0.5e1;]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:6) = [0.15e2/0.4e1 -0.77e2/0.6e1 0.107e3/0.6e1 -13 0.61e2/0.12e2 -0.5e1/0.6e1;]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:6) = [-0.17e2/0.4e1 0.71e2/0.4e1 -0.59e2/0.2e1 0.49e2/0.2e1 -0.41e2/0.4e1 0.7e1/0.4e1;]/h^3; + d3_r = -rot90(d3_l, 2); + + + % Fourth derivative, 1th order accurate at first 8 boundary points + stencil = [7/240, -2/5, 169/60, -122/15, 91/8, -122/15, 169/60, -2/5, 7/240]; + diags = -4:4; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.1399708478939e13/0.263487168000e12 -0.13482796013041e14/0.834376032000e12 0.344344095859e12/0.17565811200e11 -0.3166261424681e13/0.250312809600e12 0.1508605165681e13/0.333750412800e12 -0.486270829441e12/0.834376032000e12 -0.221976356359e12/0.5006256192000e13; + -0.13482796013041e14/0.834376032000e12 0.7260475818391e13/0.139062672000e12 -0.27224036353e11/0.406022400e9 0.1847477458951e13/0.41718801600e11 -0.848984558161e12/0.55625068800e11 0.247494925991e12/0.139062672000e12 0.165585445559e12/0.834376032000e12; + 0.344344095859e12/0.17565811200e11 -0.27224036353e11/0.406022400e9 0.2044938640393e13/0.22250027520e11 -0.1071086785417e13/0.16687520640e11 0.502199537033e12/0.22250027520e11 -0.143589154441e12/0.55625068800e11 -0.88181965559e11/0.333750412800e12; + -0.3166261424681e13/0.250312809600e12 0.1847477458951e13/0.41718801600e11 -0.1071086785417e13/0.16687520640e11 0.628860435593e12/0.12515640480e11 -0.73736245829e11/0.3337504128e10 0.195760572271e12/0.41718801600e11 -0.81156046361e11/0.250312809600e12; + 0.1508605165681e13/0.333750412800e12 -0.848984558161e12/0.55625068800e11 0.502199537033e12/0.22250027520e11 -0.73736245829e11/0.3337504128e10 0.76725285869e11/0.4450005504e10 -0.3912429433e10/0.406022400e9 0.53227370659e11/0.17565811200e11; + -0.486270829441e12/0.834376032000e12 0.247494925991e12/0.139062672000e12 -0.143589154441e12/0.55625068800e11 0.195760572271e12/0.41718801600e11 -0.3912429433e10/0.406022400e9 0.1699707221791e13/0.139062672000e12 -0.6959018412841e13/0.834376032000e12; + -0.221976356359e12/0.5006256192000e13 0.165585445559e12/0.834376032000e12 -0.88181965559e11/0.333750412800e12 -0.81156046361e11/0.250312809600e12 0.53227370659e11/0.17565811200e11 -0.6959018412841e13/0.834376032000e12 0.3012195053939e13/0.263487168000e12; + ]; + + M4(1:7,1:7) = M4_U; + M4(m-6:m,m-6:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_6_min_boundary_points.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_6_min_boundary_points.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,74 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_6_min_boundary_points(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 6:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% Extension to variable koeff %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + % H?r med 6 RP ist?llet f?r 8 f?r D4 operatorn, dock samma randderivator + + BP = 6; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:6) = [13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, 43801/43200]; + Hv(m-5:m) = rot90(Hv(1:6),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:5) = [-25/12, 4, -3, 4/3, -1/4]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:5) = [0.35e2/0.12e2 -0.26e2/0.3e1 0.19e2/0.2e1 -0.14e2/0.3e1 0.11e2/0.12e2;]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:5) = [-0.5e1/0.2e1 9 -12 7 -0.3e1/0.2e1;]/h^3; + d3_r = -rot90(d3_l, 2); + + + % Fourth derivative, 1th order accurate at first 8 boundary points (still + % yield 5th order convergence if stable: for example u_tt=-u_xxxx + + stencil = [7/240, -2/5, 169/60, -122/15, 91/8, -122/15, 169/60, -2/5, 7/240]; + diags = -4:4; + M4 = stripeMatrix(stencil, diags, m); + + M4_U=[ + 0.3504379e7/0.907200e6 -0.4613983e7/0.453600e6 0.4260437e7/0.453600e6 -0.418577e6/0.113400e6 0.524579e6/0.907200e6 0.535e3/0.18144e5; + -0.4613983e7/0.453600e6 0.5186159e7/0.181440e6 -0.81121e5/0.2835e4 0.218845e6/0.18144e5 -0.159169e6/0.90720e5 -0.94669e5/0.907200e6; + 0.4260437e7/0.453600e6 -0.81121e5/0.2835e4 0.147695e6/0.4536e4 -0.384457e6/0.22680e5 0.339653e6/0.90720e5 -0.18233e5/0.113400e6; + -0.418577e6/0.113400e6 0.218845e6/0.18144e5 -0.384457e6/0.22680e5 0.65207e5/0.4536e4 -0.22762e5/0.2835e4 0.1181753e7/0.453600e6; + 0.524579e6/0.907200e6 -0.159169e6/0.90720e5 0.339653e6/0.90720e5 -0.22762e5/0.2835e4 0.2006171e7/0.181440e6 -0.3647647e7/0.453600e6; + 0.535e3/0.18144e5 -0.94669e5/0.907200e6 -0.18233e5/0.113400e6 0.1181753e7/0.453600e6 -0.3647647e7/0.453600e6 0.10305271e8/0.907200e6; + ]; + + M4(1:6,1:6) = M4_U; + M4(m-5:m,m-5:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_8_higher_boundary_order.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_8_higher_boundary_order.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,76 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_8_higher_boundary_order(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 8:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %This is 3rd order accurate at the boundary. Not same norm as D1 operator + + BP = 8; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:8) = [0.7488203e7/0.25401600e8, 0.5539027e7/0.3628800e7, 0.308923e6/0.1209600e7, 0.1307491e7/0.725760e6, 0.59407e5/0.145152e6, 0.1548947e7/0.1209600e7, 0.3347963e7/0.3628800e7, 0.25641187e8/0.25401600e8]; + Hv(m-7:m) = rot90(Hv(1:8),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:7) = [-0.49e2/0.20e2 6 -0.15e2/0.2e1 0.20e2/0.3e1 -0.15e2/0.4e1 0.6e1/0.5e1 -0.1e1/0.6e1]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:7) = [0.203e3/0.45e2 -0.87e2/0.5e1 0.117e3/0.4e1 -0.254e3/0.9e1 0.33e2/0.2e1 -0.27e2/0.5e1 0.137e3/0.180e3]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:7) = [-0.49e2/0.8e1 29 -0.461e3/0.8e1 62 -0.307e3/0.8e1 13 -0.15e2/0.8e1]/h^3; + d3_r = -rot90(d3_l, 2); + + + + % Fourth derivative, 1th order accurate at first 8 boundary points (still + % yield 5th order convergence if stable: for example u_tt = -u_xxxx + + stencil = [-0.41e2/0.7560e4, 0.1261e4/0.15120e5, -0.541e3/0.840e3, 0.4369e4/0.1260e4, -0.1669e4/0.180e3, 0.1529e4/0.120e3, -0.1669e4/0.180e3, 0.4369e4/0.1260e4, -0.541e3/0.840e3, 0.1261e4/0.15120e5,-0.41e2/0.7560e4]; + diags = -5:5; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.1031569831e10/0.155675520e9 -0.32874237931e11/0.1452971520e10 0.3069551773e10/0.90810720e8 -0.658395212131e12/0.21794572800e11 0.31068454007e11/0.1816214400e10 -0.39244130657e11/0.7264857600e10 0.1857767503e10/0.2724321600e10 0.1009939e7/0.49420800e8; + -0.32874237931e11/0.1452971520e10 0.12799022387e11/0.155675520e9 -0.134456503627e12/0.1037836800e10 0.15366749479e11/0.129729600e9 -0.207640325549e12/0.3113510400e10 0.5396424073e10/0.259459200e9 -0.858079351e9/0.345945600e9 -0.19806607e8/0.170270100e9; + 0.3069551773e10/0.90810720e8 -0.134456503627e12/0.1037836800e10 0.6202056779e10/0.28828800e8 -0.210970327081e12/0.1037836800e10 0.2127730129e10/0.18532800e8 -0.4048692749e10/0.115315200e9 0.1025943959e10/0.259459200e9 0.71054663e8/0.290594304e9; + -0.658395212131e12/0.21794572800e11 0.15366749479e11/0.129729600e9 -0.210970327081e12/0.1037836800e10 0.31025293213e11/0.155675520e9 -0.1147729001e10/0.9884160e7 0.1178067773e10/0.32432400e8 -0.13487255581e11/0.3113510400e10 -0.231082547e9/0.1816214400e10; + 0.31068454007e11/0.1816214400e10 -0.207640325549e12/0.3113510400e10 0.2127730129e10/0.18532800e8 -0.1147729001e10/0.9884160e7 0.11524865123e11/0.155675520e9 -0.29754506009e11/0.1037836800e10 0.14231221e8/0.2316600e7 -0.15030629699e11/0.21794572800e11; + -0.39244130657e11/0.7264857600e10 0.5396424073e10/0.259459200e9 -0.4048692749e10/0.115315200e9 0.1178067773e10/0.32432400e8 -0.29754506009e11/0.1037836800e10 0.572247737e9/0.28828800e8 -0.11322059051e11/0.1037836800e10 0.3345834083e10/0.908107200e9; + 0.1857767503e10/0.2724321600e10 -0.858079351e9/0.345945600e9 0.1025943959e10/0.259459200e9 -0.13487255581e11/0.3113510400e10 0.14231221e8/0.2316600e7 -0.11322059051e11/0.1037836800e10 0.10478882597e11/0.778377600e9 -0.68446325191e11/0.7264857600e10; + 0.1009939e7/0.49420800e8 -0.19806607e8/0.170270100e9 0.71054663e8/0.290594304e9 -0.231082547e9/0.1816214400e10 -0.15030629699e11/0.21794572800e11 0.3345834083e10/0.908107200e9 -0.68446325191e11/0.7264857600e10 0.9944747557e10/0.778377600e9; + ]; + + M4(1:8,1:8) = M4_U; + M4(m-7:m,m-7:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_lonely_8_min_boundary_points.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_lonely_8_min_boundary_points.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,73 @@ +function [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_8_min_boundary_points(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 8:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + BP = 8; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:8) = [1498139/5080320, 1107307/725760, 20761/80640, 1304999/725760, 299527/725760, 103097/80640, 670091/725760, 5127739/5080320]; + Hv(m-7:m) = rot90(Hv(1:8),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:6) = [-0.137e3/0.60e2 5 -5 0.10e2/0.3e1 -0.5e1/0.4e1 0.1e1/0.5e1;]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:6) = [0.15e2/0.4e1 -0.77e2/0.6e1 0.107e3/0.6e1 -13 0.61e2/0.12e2 -0.5e1/0.6e1;]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:6) = [-0.17e2/0.4e1 0.71e2/0.4e1 -0.59e2/0.2e1 0.49e2/0.2e1 -0.41e2/0.4e1 0.7e1/0.4e1;]/h^3; + d3_r = -rot90(d3_l, 2); + + + % Fourth derivative, 1th order accurate at first 8 boundary points + + stencil = [-0.41e2/0.7560e4, 0.1261e4/0.15120e5,-0.541e3/0.840e3,0.4369e4/0.1260e4,-0.1669e4/0.180e3,0.1529e4/0.120e3,-0.1669e4/0.180e3,0.4369e4/0.1260e4,-0.541e3/0.840e3, 0.1261e4/0.15120e5,-0.41e2/0.7560e4]; + diags = -5:5; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.151705142321e12/0.29189160000e11 -0.25643455801727e14/0.1634592960000e13 0.286417898677e12/0.15135120000e11 -0.4038072020317e13/0.326918592000e12 0.96455968907e11/0.20432412000e11 -0.151076916769e12/0.181621440000e12 0.14511526363e11/0.408648240000e12 -0.196663079e9/0.33359040000e11; + -0.25643455801727e14/0.1634592960000e13 0.735383382473e12/0.14594580000e11 -0.5035391734409e13/0.77837760000e11 0.20392440917e11/0.467026560e9 -0.109540902413e12/0.6671808000e10 0.2488686539e10/0.884520000e9 -0.2798067539e10/0.33359040000e11 0.6433463591e10/0.408648240000e12; + 0.286417898677e12/0.15135120000e11 -0.5035391734409e13/0.77837760000e11 0.145019791981e12/0.1621620000e10 -0.333577111061e12/0.5189184000e10 0.18928722391e11/0.778377600e9 -0.93081704557e11/0.25945920000e11 -0.372660319e9/0.3243240000e10 0.2861399869e10/0.544864320000e12; + -0.4038072020317e13/0.326918592000e12 0.20392440917e11/0.467026560e9 -0.333577111061e12/0.5189184000e10 0.59368471277e11/0.1167566400e10 -0.201168708569e12/0.9340531200e10 0.1492314487e10/0.432432000e9 0.1911896257e10/0.9340531200e10 0.24383341e8/0.2554051500e10; + 0.96455968907e11/0.20432412000e11 -0.109540902413e12/0.6671808000e10 0.18928722391e11/0.778377600e9 -0.201168708569e12/0.9340531200e10 0.1451230301e10/0.106142400e9 -0.103548247007e12/0.15567552000e11 0.27808437809e11/0.11675664000e11 -0.36870830713e11/0.65383718400e11; + -0.151076916769e12/0.181621440000e12 0.2488686539e10/0.884520000e9 -0.93081704557e11/0.25945920000e11 0.1492314487e10/0.432432000e9 -0.103548247007e12/0.15567552000e11 0.1229498243e10/0.115830000e9 -0.32222519717e11/0.3706560000e10 0.470092704233e12/0.136216080000e12; + 0.14511526363e11/0.408648240000e12 -0.2798067539e10/0.33359040000e11 -0.372660319e9/0.3243240000e10 0.1911896257e10/0.9340531200e10 0.27808437809e11/0.11675664000e11 -0.32222519717e11/0.3706560000e10 0.11547819313e11/0.912161250e9 -0.15187033999199e14/0.1634592960000e13; + -0.196663079e9/0.33359040000e11 0.6433463591e10/0.408648240000e12 0.2861399869e10/0.544864320000e12 0.24383341e8/0.2554051500e10 -0.36870830713e11/0.65383718400e11 0.470092704233e12/0.136216080000e12 -0.15187033999199e14/0.1634592960000e13 0.33832994693e11/0.2653560000e10; + ]; + + M4(1:8,1:8) = M4_U; + M4(m-7:m,m-7:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_variable_2.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_variable_2.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,89 @@ +% Returns D2 as a function handle +function [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_2(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 4:de ordn. SBP Finita differens %%% + %%% operatorer framtagna av Ken Mattsson %%% + %%% %%% + %%% 6 randpunkter, diagonal norm %%% + %%% %%% + %%% Datum: 2013-11-11 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + BP = 2; + if(m < 2*BP) + error('Operator requires at least %d grid points', 2*BP); + end + + % Norm + Hv = ones(m,1); + Hv(1) = 1/2; + Hv(m) = 1/2; + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:3) = 1/h*[-3/2 2 -1/2]; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:3) = 1/h^2*[1 -2 1]; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:4) = 1/h^3*[-1 3 -3 1]; + d3_r = -rot90(d3_l, 2); + + + % First derivative SBP operator, 1st order accurate at first 6 boundary points + stencil = [-1/2, 0, 1/2]; + diags = [-1 0 1]; + Q = stripeMatrix(stencil, diags, m); + + D1 = HI*(Q - 1/2*e_l*e_l' + 1/2*e_r*e_r'); + + % Second derivative, 1st order accurate at first boundary points + M = sparse(m,m); + + scheme_width = 3; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + Mm1 = -c(r-1)/2 - c(r)/2; + M0 = c(r-1)/2 + c(r) + c(r+1)/2; + Mp1 = -c(r)/2 - c(r+1)/2; + + M(r,:) = spdiags([Mm1 M0 Mp1],0:2*scheme_radius,length(r),m); + + M(1:2,1:2) = [c(1)/2 + c(2)/2 -c(1)/2 - c(2)/2; -c(1)/2 - c(2)/2 c(1)/2 + c(2) + c(3)/2;]; + M(m-1:m,m-1:m) = [c(m-2)/2 + c(m-1) + c(m)/2 -c(m-1)/2 - c(m)/2; -c(m-1)/2 - c(m)/2 c(m-1)/2 + c(m)/2;]; + M = 1/h*M; + + D2 = HI*(-M - c(1)*e_l*d1_l' + c(m)*e_r*d1_r'); + end + D2 = @D2_fun; + + % Fourth derivative, 0th order accurate at first 6 boundary points + stencil = [1, -4, 6, -4, 1]; + diags = -2:2; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.13e2/0.10e2 -0.12e2/0.5e1 0.9e1/0.10e2 0.1e1/0.5e1; + -0.12e2/0.5e1 0.26e2/0.5e1 -0.16e2/0.5e1 0.2e1/0.5e1; + 0.9e1/0.10e2 -0.16e2/0.5e1 0.47e2/0.10e2 -0.17e2/0.5e1; + 0.1e1/0.5e1 0.2e1/0.5e1 -0.17e2/0.5e1 0.29e2/0.5e1; + ]; + + M4(1:4,1:4) = M4_U; + M4(m-3:m,m-3:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_variable_4.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_variable_4.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,171 @@ +function [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_4(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 4:de ordn. SBP Finita differens %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + BP = 6; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:4) = [17/48 59/48 43/48 49/48]; + Hv(m-3:m) = rot90(Hv(1:4),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:4) = 1/h*[-11/6 3 -3/2 1/3]; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:4) = 1/h^2*[2 -5 4 -1]; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:4) = 1/h^3*[-1 3 -3 1]; + d3_r = -rot90(d3_l, 2); + + + % First derivative SBP operator, + stencil = [1/12 -2/3 0 2/3 -1/12]; + diags = -2:2; + + Q_U = [ + 0 0.59e2/0.96e2 -0.1e1/0.12e2 -0.1e1/0.32e2; + -0.59e2/0.96e2 0 0.59e2/0.96e2 0; + 0.1e1/0.12e2 -0.59e2/0.96e2 0 0.59e2/0.96e2; + 0.1e1/0.32e2 0 -0.59e2/0.96e2 0; + ]; + + Q = stripeMatrix(stencil, diags, m); + Q(1:4,1:4)=Q_U; + Q(m-3:m,m-3:m) = -rot90(Q_U, 2); + + D1 = HI*(Q - 1/2*e_l*e_l' + 1/2*e_r*e_r'); + + + % Second derivative + M = sparse(m,m); + + scheme_width = 5; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + Mm2 = c(r-2)/0.8e1 - c(r-1)/0.6e1 + c(r) /0.8e1 ; + Mm1 = -c(r-2)/0.6e1 - c(r-1)/0.2e1 - c(r) /0.2e1 - c(r+1)/0.6e1 ; + M0 = c(r-2)/2.4e1 + c(r-1)/1.2e0 + c(r)*0.3/0.4 + c(r+1)/1.2e0 + c(r+2)/2.4e1; + Mp1 = -c(r-1)/0.6e1 - c(r) /0.2e1 - c(r+1)/0.2e1 - c(r+2)/0.6e1; + Mp2 = c(r) /0.8e1 - c(r+1)/0.6e1 + c(r+2)/0.8e1; + + M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m); + + M(1:6,1:6) = [ + 0.12e2/0.17e2 * c(1) + 0.59e2/0.192e3 * c(2) + 0.27010400129e11/0.345067064608e12 * c(3) + 0.69462376031e11/0.2070402387648e13 * c(4) -0.59e2/0.68e2 * c(1) - 0.6025413881e10/0.21126554976e11 * c(3) - 0.537416663e9/0.7042184992e10 * c(4) 0.2e1/0.17e2 * c(1) - 0.59e2/0.192e3 * c(2) + 0.213318005e9/0.16049630912e11 * c(4) + 0.2083938599e10/0.8024815456e10 * c(3) 0.3e1/0.68e2 * c(1) - 0.1244724001e10/0.21126554976e11 * c(3) + 0.752806667e9/0.21126554976e11 * c(4) 0.49579087e8/0.10149031312e11 * c(3) - 0.49579087e8/0.10149031312e11 * c(4) -c(4)/0.784e3 + c(3)/0.784e3; + -0.59e2/0.68e2 * c(1) - 0.6025413881e10/0.21126554976e11 * c(3) - 0.537416663e9/0.7042184992e10 * c(4) 0.3481e4/0.3264e4 * c(1) + 0.9258282831623875e16/0.7669235228057664e16 * c(3) + 0.236024329996203e15/0.1278205871342944e16 * c(4) -0.59e2/0.408e3 * c(1) - 0.29294615794607e14/0.29725717938208e14 * c(3) - 0.2944673881023e13/0.29725717938208e14 * c(4) -0.59e2/0.1088e4 * c(1) + 0.260297319232891e15/0.2556411742685888e16 * c(3) - 0.60834186813841e14/0.1278205871342944e16 * c(4) -0.1328188692663e13/0.37594290333616e14 * c(3) + 0.1328188692663e13/0.37594290333616e14 * c(4) -0.8673e4/0.2904112e7 * c(3) + 0.8673e4/0.2904112e7 * c(4); + 0.2e1/0.17e2 * c(1) - 0.59e2/0.192e3 * c(2) + 0.213318005e9/0.16049630912e11 * c(4) + 0.2083938599e10/0.8024815456e10 * c(3) -0.59e2/0.408e3 * c(1) - 0.29294615794607e14/0.29725717938208e14 * c(3) - 0.2944673881023e13/0.29725717938208e14 * c(4) c(1)/0.51e2 + 0.59e2/0.192e3 * c(2) + 0.13777050223300597e17/0.26218083221499456e17 * c(4) + 0.564461e6/0.13384296e8 * c(5) + 0.378288882302546512209e21/0.270764341349677687456e21 * c(3) c(1)/0.136e3 - 0.125059e6/0.743572e6 * c(5) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(3) - 0.17220493277981e14/0.89177153814624e14 * c(4) -0.10532412077335e14/0.42840005263888e14 * c(4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(3) + 0.564461e6/0.4461432e7 * c(5) -0.960119e6/0.1280713392e10 * c(4) - 0.3391e4/0.6692148e7 * c(5) + 0.33235054191e11/0.26452850508784e14 * c(3); + 0.3e1/0.68e2 * c(1) - 0.1244724001e10/0.21126554976e11 * c(3) + 0.752806667e9/0.21126554976e11 * c(4) -0.59e2/0.1088e4 * c(1) + 0.260297319232891e15/0.2556411742685888e16 * c(3) - 0.60834186813841e14/0.1278205871342944e16 * c(4) c(1)/0.136e3 - 0.125059e6/0.743572e6 * c(5) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(3) - 0.17220493277981e14/0.89177153814624e14 * c(4) 0.3e1/0.1088e4 * c(1) + 0.507284006600757858213e21/0.475219048083107777984e21 * c(3) + 0.1869103e7/0.2230716e7 * c(5) + c(6)/0.24e2 + 0.1950062198436997e16/0.3834617614028832e16 * c(4) -0.4959271814984644613e19/0.20965546238960637264e20 * c(3) - c(6)/0.6e1 - 0.15998714909649e14/0.37594290333616e14 * c(4) - 0.375177e6/0.743572e6 * c(5) -0.368395e6/0.2230716e7 * c(5) + 0.752806667e9/0.539854092016e12 * c(3) + 0.1063649e7/0.8712336e7 * c(4) + c(6)/0.8e1; + 0.49579087e8/0.10149031312e11 * c(3) - 0.49579087e8/0.10149031312e11 * c(4) -0.1328188692663e13/0.37594290333616e14 * c(3) + 0.1328188692663e13/0.37594290333616e14 * c(4) -0.10532412077335e14/0.42840005263888e14 * c(4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(3) + 0.564461e6/0.4461432e7 * c(5) -0.4959271814984644613e19/0.20965546238960637264e20 * c(3) - c(6)/0.6e1 - 0.15998714909649e14/0.37594290333616e14 * c(4) - 0.375177e6/0.743572e6 * c(5) 0.8386761355510099813e19/0.128413970713633903242e21 * c(3) + 0.2224717261773437e16/0.2763180339520776e16 * c(4) + 0.5e1/0.6e1 * c(6) + c(7)/0.24e2 + 0.280535e6/0.371786e6 * c(5) -0.35039615e8/0.213452232e9 * c(4) - c(7)/0.6e1 - 0.13091810925e11/0.13226425254392e14 * c(3) - 0.1118749e7/0.2230716e7 * c(5) - c(6)/0.2e1; + -c(4)/0.784e3 + c(3)/0.784e3 -0.8673e4/0.2904112e7 * c(3) + 0.8673e4/0.2904112e7 * c(4) -0.960119e6/0.1280713392e10 * c(4) - 0.3391e4/0.6692148e7 * c(5) + 0.33235054191e11/0.26452850508784e14 * c(3) -0.368395e6/0.2230716e7 * c(5) + 0.752806667e9/0.539854092016e12 * c(3) + 0.1063649e7/0.8712336e7 * c(4) + c(6)/0.8e1 -0.35039615e8/0.213452232e9 * c(4) - c(7)/0.6e1 - 0.13091810925e11/0.13226425254392e14 * c(3) - 0.1118749e7/0.2230716e7 * c(5) - c(6)/0.2e1 0.3290636e7/0.80044587e8 * c(4) + 0.5580181e7/0.6692148e7 * c(5) + 0.5e1/0.6e1 * c(7) + c(8)/0.24e2 + 0.660204843e9/0.13226425254392e14 * c(3) + 0.3e1/0.4e1 * c(6); + ]; + + M(m-5:m,m-5:m) = [ + c(m-7)/0.24e2 + 0.5e1/0.6e1 * c(m-6) + 0.5580181e7/0.6692148e7 * c(m-4) + 0.4887707739997e13/0.119037827289528e15 * c(m-3) + 0.3e1/0.4e1 * c(m-5) + 0.660204843e9/0.13226425254392e14 * c(m-2) + 0.660204843e9/0.13226425254392e14 * c(m-1) -c(m-6)/0.6e1 - 0.1618585929605e13/0.9919818940794e13 * c(m-3) - c(m-5)/0.2e1 - 0.1118749e7/0.2230716e7 * c(m-4) - 0.13091810925e11/0.13226425254392e14 * c(m-2) - 0.13091810925e11/0.13226425254392e14 * c(m-1) -0.368395e6/0.2230716e7 * c(m-4) + c(m-5)/0.8e1 + 0.48866620889e11/0.404890569012e12 * c(m-3) + 0.752806667e9/0.539854092016e12 * c(m-2) + 0.752806667e9/0.539854092016e12 * c(m-1) -0.3391e4/0.6692148e7 * c(m-4) - 0.238797444493e12/0.119037827289528e15 * c(m-3) + 0.33235054191e11/0.26452850508784e14 * c(m-2) + 0.33235054191e11/0.26452850508784e14 * c(m-1) -0.8673e4/0.2904112e7 * c(m-2) - 0.8673e4/0.2904112e7 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -c(m-3)/0.392e3 + c(m-2)/0.784e3 + c(m-1)/0.784e3; + -c(m-6)/0.6e1 - 0.1618585929605e13/0.9919818940794e13 * c(m-3) - c(m-5)/0.2e1 - 0.1118749e7/0.2230716e7 * c(m-4) - 0.13091810925e11/0.13226425254392e14 * c(m-2) - 0.13091810925e11/0.13226425254392e14 * c(m-1) c(m-6)/0.24e2 + 0.5e1/0.6e1 * c(m-5) + 0.3896014498639e13/0.4959909470397e13 * c(m-3) + 0.8386761355510099813e19/0.128413970713633903242e21 * c(m-2) + 0.280535e6/0.371786e6 * c(m-4) + 0.3360696339136261875e19/0.171218627618178537656e21 * c(m-1) -c(m-5)/0.6e1 - 0.4959271814984644613e19/0.20965546238960637264e20 * c(m-2) - 0.375177e6/0.743572e6 * c(m-4) - 0.13425842714e11/0.33740880751e11 * c(m-3) - 0.193247108773400725e18/0.6988515412986879088e19 * c(m-1) -0.365281640980e12/0.1653303156799e13 * c(m-3) + 0.564461e6/0.4461432e7 * c(m-4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18/0.7963657098519931984e19 * c(m-1) -0.1328188692663e13/0.37594290333616e14 * c(m-2) + 0.2226377963775e13/0.37594290333616e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) c(m-3)/0.49e2 + 0.49579087e8/0.10149031312e11 * c(m-2) - 0.256702175e9/0.10149031312e11 * c(m-1); + -0.368395e6/0.2230716e7 * c(m-4) + c(m-5)/0.8e1 + 0.48866620889e11/0.404890569012e12 * c(m-3) + 0.752806667e9/0.539854092016e12 * c(m-2) + 0.752806667e9/0.539854092016e12 * c(m-1) -c(m-5)/0.6e1 - 0.4959271814984644613e19/0.20965546238960637264e20 * c(m-2) - 0.375177e6/0.743572e6 * c(m-4) - 0.13425842714e11/0.33740880751e11 * c(m-3) - 0.193247108773400725e18/0.6988515412986879088e19 * c(m-1) c(m-5)/0.24e2 + 0.1869103e7/0.2230716e7 * c(m-4) + 0.507284006600757858213e21/0.475219048083107777984e21 * c(m-2) + 0.3e1/0.1088e4 * c(m) + 0.31688435395e11/0.67481761502e11 * c(m-3) + 0.27769176016102795561e20/0.712828572124661666976e21 * c(m-1) -0.125059e6/0.743572e6 * c(m-4) + c(m)/0.136e3 - 0.23099342648e11/0.101222642253e12 * c(m-3) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18/0.5525802884687299744e19 * c(m-1) 0.260297319232891e15/0.2556411742685888e16 * c(m-2) - 0.59e2/0.1088e4 * c(m) - 0.106641839640553e15/0.1278205871342944e16 * c(m-1) + 0.26019e5/0.726028e6 * c(m-3) -0.1244724001e10/0.21126554976e11 * c(m-2) + 0.3e1/0.68e2 * c(m) + 0.752806667e9/0.21126554976e11 * c(m-1); + -0.3391e4/0.6692148e7 * c(m-4) - 0.238797444493e12/0.119037827289528e15 * c(m-3) + 0.33235054191e11/0.26452850508784e14 * c(m-2) + 0.33235054191e11/0.26452850508784e14 * c(m-1) -0.365281640980e12/0.1653303156799e13 * c(m-3) + 0.564461e6/0.4461432e7 * c(m-4) + 0.1613976761032884305e19/0.7963657098519931984e19 * c(m-2) - 0.198407225513315475e18/0.7963657098519931984e19 * c(m-1) -0.125059e6/0.743572e6 * c(m-4) + c(m)/0.136e3 - 0.23099342648e11/0.101222642253e12 * c(m-3) - 0.4836340090442187227e19/0.5525802884687299744e19 * c(m-2) + 0.193950157930938693e18/0.5525802884687299744e19 * c(m-1) 0.564461e6/0.13384296e8 * c(m-4) + 0.470299699916357e15/0.952302618316224e15 * c(m-3) + 0.550597048646198778781e21/0.1624586048098066124736e22 * c(m-1) + c(m)/0.51e2 + 0.378288882302546512209e21/0.270764341349677687456e21 * c(m-2) -0.59e2/0.408e3 * c(m) - 0.29294615794607e14/0.29725717938208e14 * c(m-2) - 0.2234477713167e13/0.29725717938208e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) -0.59e2/0.3136e4 * c(m-3) - 0.13249937023e11/0.48148892736e11 * c(m-1) + 0.2e1/0.17e2 * c(m) + 0.2083938599e10/0.8024815456e10 * c(m-2); + -0.8673e4/0.2904112e7 * c(m-2) - 0.8673e4/0.2904112e7 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -0.1328188692663e13/0.37594290333616e14 * c(m-2) + 0.2226377963775e13/0.37594290333616e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) 0.260297319232891e15/0.2556411742685888e16 * c(m-2) - 0.59e2/0.1088e4 * c(m) - 0.106641839640553e15/0.1278205871342944e16 * c(m-1) + 0.26019e5/0.726028e6 * c(m-3) -0.59e2/0.408e3 * c(m) - 0.29294615794607e14/0.29725717938208e14 * c(m-2) - 0.2234477713167e13/0.29725717938208e14 * c(m-1) - 0.8673e4/0.363014e6 * c(m-3) 0.9258282831623875e16/0.7669235228057664e16 * c(m-2) + 0.3481e4/0.3264e4 * c(m) + 0.228389721191751e15/0.1278205871342944e16 * c(m-1) + 0.8673e4/0.1452056e7 * c(m-3) -0.6025413881e10/0.21126554976e11 * c(m-2) - 0.59e2/0.68e2 * c(m) - 0.537416663e9/0.7042184992e10 * c(m-1); + -c(m-3)/0.392e3 + c(m-2)/0.784e3 + c(m-1)/0.784e3 c(m-3)/0.49e2 + 0.49579087e8/0.10149031312e11 * c(m-2) - 0.256702175e9/0.10149031312e11 * c(m-1) -0.1244724001e10/0.21126554976e11 * c(m-2) + 0.3e1/0.68e2 * c(m) + 0.752806667e9/0.21126554976e11 * c(m-1) -0.59e2/0.3136e4 * c(m-3) - 0.13249937023e11/0.48148892736e11 * c(m-1) + 0.2e1/0.17e2 * c(m) + 0.2083938599e10/0.8024815456e10 * c(m-2) -0.6025413881e10/0.21126554976e11 * c(m-2) - 0.59e2/0.68e2 * c(m) - 0.537416663e9/0.7042184992e10 * c(m-1) 0.3e1/0.3136e4 * c(m-3) + 0.27010400129e11/0.345067064608e12 * c(m-2) + 0.234566387291e12/0.690134129216e12 * c(m-1) + 0.12e2/0.17e2 * c(m); + ]; + + M = M/h; + D2 = HI*(-M - c(1)*e_l*d1_l' + c(m)*e_r*d1_r'); + end + D2 = @D2_fun; + + + % Fourth derivative + stencil = [-1/6,2,-13/2, 28/3,-13/2,2,-1/6]; + diags = -3:3; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.5762947e7/0.2316384e7 -0.6374287e7/0.1158192e7 0.573947e6/0.165456e6 -0.124637e6/0.289548e6 0.67979e5/0.2316384e7 -0.60257e5/0.1158192e7; + -0.6374287e7/0.1158192e7 0.30392389e8/0.2316384e7 -0.2735053e7/0.289548e6 0.273109e6/0.165456e6 0.83767e5/0.1158192e7 0.245549e6/0.2316384e7; + 0.573947e6/0.165456e6 -0.2735053e7/0.289548e6 0.5266855e7/0.579096e6 -0.1099715e7/0.289548e6 0.869293e6/0.1158192e7 -0.10195e5/0.144774e6; + -0.124637e6/0.289548e6 0.273109e6/0.165456e6 -0.1099715e7/0.289548e6 0.3259225e7/0.579096e6 -0.324229e6/0.72387e5 0.1847891e7/0.1158192e7; + 0.67979e5/0.2316384e7 0.83767e5/0.1158192e7 0.869293e6/0.1158192e7 -0.324229e6/0.72387e5 0.2626501e7/0.330912e6 -0.7115491e7/0.1158192e7; + -0.60257e5/0.1158192e7 0.245549e6/0.2316384e7 -0.10195e5/0.144774e6 0.1847891e7/0.1158192e7 -0.7115491e7/0.1158192e7 0.21383077e8/0.2316384e7; + ]; + + M4(1:6,1:6) = M4_U; + M4(m-5:m,m-5:m) = rot90(M4_U, 2); + M4 = 1/h^3*M4; + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end + + +% Alternatives for variable 2nd derivative + % % ALTERNATIVES %%%%%%%%%%%%% + % % for i=4:m-3 + % % M(i,i-2:i+2)=[-c(i-1)/0.6e1 + c(i-2)/0.8e1 + c(i)/0.8e1 -c(i-2)/0.6e1 - c(i+1)/0.6e1 - c(i-1)/0.2e1 - c(i)/0.2e1 c(i-2)/0.24e2 + 0.5e1/0.6e1 * c(i-1) + 0.5e1/0.6e1 * c(i+1) + c(i+2)/0.24e2 + 0.3e1/0.4e1 * c(i) -c(i-1)/0.6e1 - c(i+2)/0.6e1 - c(i)/0.2e1 - c(i+1)/0.2e1 -c(i+1)/0.6e1 + c(i)/0.8e1 + c(i+2)/0.8e1;]; + % % end + % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % % for i=4:m-3 + % % M(i,i-2:i+2)= [ + % % c(i-2)/0.8e1 - c(i-1)/0.6e1 + c(i) /0.8e1 , + % % -c(i-2)/0.6e1 - c(i-1)/0.2e1 - c(i) /0.2e1 - c(i+1)/0.6e1 , + % % c(i-2)/2.4e1 + c(i-1)/1.2e0 + c(i) * 0.3/0.4 + c(i+1)/1.2e0 + c(i+2)/2.4e1 , + % % -c(i-1)/0.6e1 - c(i) /0.2e1 - c(i+1)/0.2e1 - c(i+2)/0.6e1 , + % % c(i) /0.8e1 - c(i+1)/0.6e1 + c(i+2)/0.8e1 , + % % ]; + % % end + % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % Mm2 = c(r-2)/0.8e1 - c(r-1)/0.6e1 + c(r) /0.8e1 ; + % Mm1 = -c(r-2)/0.6e1 - c(r-1)/0.2e1 - c(r) /0.2e1 - c(r+1)/0.6e1 ; + % M0 = c(r-2)/2.4e1 + c(r-1)/1.2e0 + c(r) * 0.3/0.4 + c(r+1)/1.2e0 + c(r+2)/2.4e1; + % Mp1 = -c(r-1)/0.6e1 - c(r) /0.2e1 - c(r+1)/0.2e1 - c(r+2)/0.6e1; + % Mp2 = c(r) /0.8e1 - c(r+1)/0.6e1 + c(r+2)/0.8e1; + + % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],0:2*scheme_radius,length(r),m); + % % M(r,:) = spdiags([Mm2 Mm1 M0 Mp1 Mp2],(-2:2)+scheme_radius,M(r,:)); % This is slower + % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % % %% Somthing is wrong here!! + % % Mm2 = c(r-2)/0.8e1 - c(r-1)/0.6e1 + c(r) /0.8e1 ; + % % Mm1 = -c(r-2)/0.6e1 - c(r-1)/0.2e1 - c(r) /0.2e1 - c(r+1)/0.6e1 ; + % % M0 = c(r-2)/2.4e1 + c(r-1)/1.2e0 + c(r) * 0.3/0.4 + c(r+1)/1.2e0 + c(r+2)/2.4e1; + % % Mp1 = -c(r-1)/0.6e1 - c(r) /0.2e1 - c(r+1)/0.2e1 - c(r+2)/0.6e1; + % % Mp2 = c(r) /0.8e1 - c(r+1)/0.6e1 + c(r+2)/0.8e1; + % % % printSize(M_diag_ind); + % % % Mdiags = [Mm2 Mm1 M0 Mp1 Mp2]; + % % % printSize(Mdiags); + % % M(M_diag_ind) = [Mm2 Mm1 M0 Mp1 Mp2]; % This is slightly faster + % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + % % Kan man skriva det som en multiplikation av en 3-dim matris? + % %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + + + + + + + +
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/+implementations/d4_variable_6.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d4_variable_6.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,158 @@ +function [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = d4_variable_6(m,h) + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + %%% 6:te ordn. SBP Finita differens %%% + %%% operatorer med diagonal norm %%% + %%% Extension to variable koeff %%% + %%% %%% + %%% H (Normen) %%% + %%% D1=H^(-1)Q (approx f?rsta derivatan) %%% + %%% D2 (approx andra derivatan) %%% + %%% D2=HI*(R+C*D*S %%% + %%% %%% + %%% R=-D1'*H*C*D1-RR %%% + %%% %%% + %%% RR ?r dissipation) %%% + %%% Dissipationen uppbyggd av D4: %%% + %%% DI=D4*B*H*D4 %%% + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + BP = 9; + if(m<2*BP) + error(['Operator requires at least ' num2str(2*BP) ' grid points']); + end + + % Norm + Hv = ones(m,1); + Hv(1:6) = [13649/43200,12013/8640,2711/4320,5359/4320,7877/8640, 43801/43200]; + Hv(m-5:m) = rot90(Hv(1:6),2); + Hv = h*Hv; + H = spdiag(Hv, 0); + HI = spdiag(1./Hv, 0); + + + % Boundary operators + e_l = sparse(m,1); + e_l(1) = 1; + e_r = rot90(e_l, 2); + + d1_l = sparse(m,1); + d1_l(1:5) = [-25/12, 4, -3, 4/3, -1/4]/h; + d1_r = -rot90(d1_l, 2); + + d2_l = sparse(m,1); + d2_l(1:5) = [0.35e2/0.12e2 -0.26e2/0.3e1 0.19e2/0.2e1 -0.14e2/0.3e1 0.11e2/0.12e2;]/h^2; + d2_r = rot90(d2_l, 2); + + d3_l = sparse(m,1); + d3_l(1:5) = [-5/2 9 -12 7 -3/2]/h^3; + d3_r = -rot90(d3_l, 2); + + + % First derivtive + x1=0.70127127127127; + + + D1=(1/60*diag(ones(m-3,1),3)-9/60*diag(ones(m-2,1),2)+45/60*diag(ones(m-1,1),1)-45/60*diag(ones(m-1,1),-1)+9/60*diag(ones(m-2,1),-2)-1/60*diag(ones(m-3,1),-3)); + + + + D1(1:6,1:9)=[-21600/13649, 43200/13649*x1-7624/40947, -172800/13649*x1+ ... + 715489/81894, 259200/13649*x1-187917/13649, -172800/13649* ... + x1+735635/81894, 43200/13649*x1-89387/40947, 0, 0, 0; ... + -8640/12013*x1+7624/180195, 0, 86400/12013*x1-57139/12013, ... + -172800/12013*x1+745733/72078, 129600/12013*x1-91715/12013, ... + -34560/12013*x1+240569/120130, 0, 0, 0; ... + 17280/2711*x1-715489/162660, -43200/2711*x1+57139/5422, 0, ... + 86400/2711*x1-176839/8133, -86400/2711*x1+242111/10844, ... + 25920/2711*x1-182261/27110, 0, 0, 0; ... + -25920/5359*x1+187917/53590, 86400/5359*x1-745733/64308, ... + -86400/5359*x1+176839/16077, 0, 43200/5359*x1-165041/32154, ... + -17280/5359*x1+710473/321540, 72/5359, 0, 0; ... + 34560/7877*x1-147127/47262, -129600/7877*x1+91715/7877, ... + 172800/7877*x1-242111/15754, -86400/7877*x1+165041/23631, ... + 0, 8640/7877*x1, -1296/7877, 144/7877, 0; ... + -43200/43801*x1+89387/131403, 172800/43801*x1-240569/87602,... + -259200/43801*x1+182261/43801, 172800/43801*x1-710473/262806, ... + -43200/43801*x1, 0, 32400/43801, -6480/43801, 720/43801]; + D1(m-5:m,m-8:m)=rot90( -D1(1:6,1:9),2); + D1=D1/h; + + + % Second derivative + M=sparse(m,m); + + scheme_width = 7; + scheme_radius = (scheme_width-1)/2; + r = (1+scheme_radius):(m-scheme_radius); + + function D2 = D2_fun(c) + + Mm3 = c(r-2)/0.40e2 + c(r-1)/0.40e2 - 0.11e2/0.360e3 * c(r-3) - 0.11e2/0.360e3 * c(r); + Mm2 = c(r-3)/0.20e2 - 0.3e1/0.10e2 * c(r-1) + c(r+1)/0.20e2 + 0.7e1/0.40e2 * c(r) + 0.7e1/0.40e2 * c(r-2); + Mm1 = -c(r-3)/0.40e2 - 0.3e1/0.10e2 * c(r-2) - 0.3e1/0.10e2 * c(r+1) - c(r+2)/0.40e2 - 0.17e2/0.40e2 * c(r) - 0.17e2/0.40e2 * c(r-1); + M0 = c(r-3)/0.180e3 + c(r-2)/0.8e1 + 0.19e2/0.20e2 * c(r-1) + 0.19e2/0.20e2 * c(r+1) + c(r+2)/0.8e1 + c(r+3)/0.180e3 + 0.101e3/0.180e3 * c(r); + Mp1 = -c(r-2)/0.40e2 - 0.3e1/0.10e2 * c(r-1) - 0.3e1/0.10e2 * c(r+2) - c(r+3)/0.40e2 - 0.17e2/0.40e2 * c(r) - 0.17e2/0.40e2 * c(r+1); + Mp2 = c(r-1)/0.20e2 - 0.3e1/0.10e2 * c(r+1) + c(r+3)/0.20e2 + 0.7e1/0.40e2 * c(r) + 0.7e1/0.40e2 * c(r+2); + Mp3 = c(r+1)/0.40e2 + c(r+2)/0.40e2 - 0.11e2/0.360e3 * c(r) - 0.11e2/0.360e3 * c(r+3); + + M(r,:) = spdiags([Mm3 Mm2 Mm1 M0 Mp1 Mp2 Mp3],0:2*scheme_radius,length(r),m); + + + M(1:9,1:9)=[ + 0.7912667594695582093926295e0 * c(1) + 0.2968472090638000742888467e0 * c(2) + 0.3185519088796429015220016e-2 * c(3) + 0.1632404042590951953384672e-1 * c(4) + 0.3160302244094415087693968e-1 * c(5) + 0.3167964748016105299646518e-1 * c(6) + 0.3148577733947253920469418e-1 * c(7) -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7); + -0.1016689339350338144430605e1 * c(1) - 0.2845627370491611369031341e-1 * c(3) - 0.4128029838349298819825156e-1 * c(4) - 0.1392281451620140507549866e0 * c(5) - 0.1195777325611201766551392e0 * c(6) - 0.1194267756529333410855186e0 * c(7) 0.1306332157111667628555907e1 * c(1) + 0.2542001760457345743492403e0 * c(3) + 0.1043897828092562609502636e0 * c(4) + 0.6672328021032112950919876e0 * c(5) + 0.4681819359722749441073885e0 * c(6) + 0.4676415410195836920069412e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7); + 0.7075642937243715046279337e-1 * c(1) - 0.1845476106024151050283847e0 * c(2) - 0.4364163147111892346990101e-1 * c(4) + 0.2432367907207732460874765e0 * c(5) + 0.1582127073537215443965653e0 * c(6) + 0.1602348578364786307613271e0 * c(7) -0.9091410269992464604926176e-1 * c(1) + 0.1103611313171476425250639e0 * c(4) - 0.1290397544997518887000350e1 * c(5) - 0.6639605248735044787146222e0 * c(6) - 0.6615974464005206184151509e0 * c(7) 0.6327161147136873807796515e-2 * c(1) + 0.1147318200715868527529827e0 * c(2) + 0.1166740554279680007487795e0 * c(4) + 0.2766610808285444037240703e1 * c(5) + 0.1070920689960817104203947e1 * c(6) + 0.1013161391032973057171717e1 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7); + 0.2251991532891353212689574e0 * c(1) - 0.1662748711097054895317080e0 * c(2) + 0.2710530961648671297733465e-1 * c(3) - 0.1916646185968439909125616e0 * c(5) - 0.7684117160199014594442072e-1 * c(6) - 0.8219586949831697575883635e-1 * c(7) -0.2893557395653431666593814e0 * c(1) - 0.2421320004064592721552708e0 * c(3) + 0.1187670255028031027693374e1 * c(5) + 0.3956598149904136332753521e0 * c(6) + 0.3860048921755800000681479e0 * c(7) 0.2013769413884797246646959e-1 * c(1) + 0.1033717994630886401730470e0 * c(2) - 0.2913221621151742724258117e1 * c(5) - 0.8755807343482262259774782e0 * c(6) - 0.6909957183488812426508351e0 * c(7) 0.6409299775987186986730499e-1 * c(1) + 0.9313657638804699948929701e-1 * c(2) + 0.2306367624634749229113646e0 * c(3) + 0.3689440308283716620260816e1 * c(5) + 0.1190550338687608873798462e1 * c(6) + 0.5912479546888856519443605e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5); + -0.5224403464202056316702078e-1 * c(1) + 0.4440063948509876221050939e-1 * c(2) - 0.1023976547309387874453988e-2 * c(3) + 0.7403484645316174090533193e-1 * c(4) + 0.1241625568998496895352046e-1 * c(6) + 0.7188652847892601282652228e-1 * c(5) + 0.1379362997104735503447960e-1 * c(7) 0.6712774475803763988977355e-1 * c(1) + 0.9147192682075630179962131e-2 * c(3) - 0.1872196143003808021730728e0 * c(4) - 0.1319358558853174530078498e0 * c(6) - 0.4871575736811911887376923e0 * c(5) - 0.1047516312275448138054418e0 * c(7) -0.4671751091575462868310238e-2 * c(1) - 0.2760353365637712827793337e-1 * c(2) - 0.1979290298620869974478871e0 * c(4) + 0.5402985338373433052255418e0 * c(6) + 0.1239177593031911077924537e1 * c(5) + 0.2628038050247358227280031e0 * c(7) -0.1486895819265604128572498e-1 * c(1) - 0.2487040599390160764166412e-1 * c(2) - 0.8712928907711754187084757e-2 * c(3) - 0.1263507837371824205693950e1 * c(6) - 0.3058317397843997326920898e0 * c(7) - 0.1470691926045802954795783e1 * c(5) 0.3449455095910233625229891e-2 * c(1) + 0.6641183499427826101618457e-2 * c(2) + 0.3291545083271862858501887e-3 * c(3) + 0.3357721707576477199985656e0 * c(4) + 0.2096413329579026439044119e1 * c(6) + 0.2317323204183126854954203e0 * c(7) + 0.6107825764368264576481962e-2 * c(8) + 0.7109125850683376695640722e0 * c(5) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5); + -0.1828896813877197352675410e-1 * c(1) + 0.9574633163221758060736592e-2 * c(2) - 0.8105784530576404277872603e-3 * c(3) - 0.7348845587775519698437916e-2 * c(4) + 0.1063601949723906997026904e-1 * c(5) - 0.1315967038382618382356495e-1 * c(6) - 0.2117936478838753524581943e-1 * c(7) 0.2349927974590068869356781e-1 * c(1) + 0.7240905383565181316381731e-2 * c(3) + 0.1858378996391679448655070e-1 * c(4) - 0.9289616133938676174345208e-1 * c(5) + 0.1223513270418807666970488e0 * c(6) + 0.1113520320436295033894092e0 * c(7) -0.1635430866921887819487473e-2 * c(1) - 0.5952475275883259619711594e-2 * c(2) + 0.1964682777744275219350831e-1 * c(4) + 0.3236640012639046600590714e0 * c(5) - 0.4659516693228870973898560e0 * c(6) - 0.2217272720941736859420432e0 * c(7) -0.5205147429855955657625694e-2 * c(1) - 0.5363098747528542488971874e-2 * c(2) - 0.6897142765790609546343709e-2 * c(3) - 0.7857524521667450101721993e0 * c(5) + 0.2291148005423734600066709e0 * c(7) + 0.9977064356292750529201981e0 * c(6) 0.1207544072304193806052558e-2 * c(1) + 0.1432116665752147607469646e-2 * c(2) + 0.2605582646183255957264249e-3 * c(3) - 0.3332941113251635390801278e-1 * c(4) - 0.2808241697385532683612407e0 * c(7) - 0.2720908083525083608370563e-1 * c(8) + 0.1045865435682921987447929e0 * c(5) - 0.1348436986667115543203552e1 * c(6) 0.4227226173449345042468960e-3 * c(1) + 0.3088241944378964404772302e-3 * c(2) + 0.2062575706647430620228133e-3 * c(3) + 0.3308343404200968256656458e-2 * c(4) + 0.5828047016405001815804837e0 * c(5) + 0.8054174220366215473556835e0 * c(7) + 0.1338363233410033443348225e0 * c(8) + 0.5555555555555555555555556e-2 * c(9) + 0.1190362071861893051132274e1 * c(6) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9); + 0.1911888563316170927411831e-2 * c(4) - 0.4068130355529149936100229e-1 * c(5) + 0.1319674981073749167009902e-1 * c(6) + 0.2557266518123783676349144e-1 * c(7) -0.4834791406446907590553793e-2 * c(4) + 0.2310683832687820403062715e0 * c(5) - 0.1080774142196007991746827e0 * c(6) - 0.1181561776427343335410349e0 * c(7) -0.5111353189352474549563559e-2 * c(4) - 0.5355878163774754346032096e0 * c(5) + 0.3328335104489738933610597e0 * c(6) + 0.2078656591178540157917135e0 * c(7) 0.6697297488067662265210608e0 * c(5) - 0.5013247356072127938999311e0 * c(6) - 0.1795161243106645437322408e0 * c(7) 0.8671038084174692625075159e-2 * c(4) + 0.1736073411355428563685818e0 * c(6) + 0.5331362125287625412555844e-1 * c(8) - 0.2424935262404526301801157e0 * c(5) + 0.1569015257678588270609004e0 * c(7) -0.8607044252686413302647675e-3 * c(4) - 0.1748074708673904989293256e0 * c(5) - 0.3132544850115050165022338e0 * c(8) - 0.2500000000000000000000000e-1 * c(9) - 0.3169166305310429271303167e0 * c(7) - 0.6691607091647929161078591e0 * c(6) 0.2239223735771599178951297e-3 * c(4) + 0.1275437785430956673825710e0 * c(5) + 0.1011699483929608164601067e1 * c(6) + 0.9698817275172575247533506e0 * c(8) + 0.1250000000000000000000000e0 * c(9) + 0.5555555555555555555555556e-2 * c(10) + 0.4823177543031281500117826e0 * c(7) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9); + 0.1559652871136785763960685e-1 * c(5) - 0.6486184157331537899459796e-2 * c(6) - 0.9110344554036319740147054e-2 * c(7) -0.8368141434403455353724691e-1 * c(5) + 0.4093499466767054661591066e-1 * c(6) + 0.4274641967636400692133626e-1 * c(7) 0.1824328174134289562208038e0 * c(5) - 0.1059816030196818445908057e0 * c(6) - 0.7645121439374711162999809e-1 * c(7) -0.2022909060111751565150958e0 * c(5) + 0.1453421858063658498587377e0 * c(6) + 0.5694872020480930665635812e-1 * c(7) -0.8631683980217122275970376e-1 * c(6) + 0.2698842360470999243492629e-1 * c(7) + 0.8098194147715651085292754e-1 * c(5) - 0.3276463639080639163926118e-1 * c(8) 0.3354661791693352108660900e-1 * c(5) - 0.3343620022386971405018586e0 * c(7) + 0.5000000000000000000000000e-1 * c(9) + 0.2169790609807602750804271e0 * c(6) + 0.1838363233410033443348225e0 * c(8) -0.3784113973033012949863031e-1 * c(5) - 0.2997556885134827361576001e0 * c(6) - 0.3000000000000000000000000e0 * c(9) - 0.2500000000000000000000000e-1 * c(10) - 0.3991486867446821178415359e0 * c(7) - 0.4382544850115050165022338e0 * c(8) 0.1230328942716804455358698e-1 * c(5) + 0.1183647529645898332481833e0 * c(6) + 0.9410511898227943334189628e0 * c(7) + 0.9500000000000000000000000e0 * c(9) + 0.1250000000000000000000000e0 * c(10) + 0.5555555555555555555555556e-2 * c(11) + 0.5699474344521144554459336e0 * c(8) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9); + 0.5593983696629863059347067e-3 * c(6) - 0.1384822535100796372263822e-2 * c(5) + 0.8254241654378100663291154e-3 * c(7) -0.3576545132696983143406173e-2 * c(6) + 0.7389399124121078682094445e-2 * c(5) - 0.3812853991424095538688273e-2 * c(7) 0.9209089963443799485648361e-2 * c(6) - 0.1591502818872493167091475e-1 * c(5) + 0.6705938225281132185266388e-2 * c(7) -0.1200429618441003833696998e-1 * c(6) - 0.4776915669385923841535432e-2 * c(7) + 0.1678121185379596217850541e-1 * c(5) 0.7462059484530855073291365e-2 * c(6) - 0.8121640361668678949573496e-3 * c(7) + 0.5522702088127090209264064e-3 * c(8) - 0.7202165657176696199260422e-2 * c(5) 0.2912518476823004642951502e-1 * c(7) + 0.2279091916474916391629437e-1 * c(8) - 0.3068985997518740530511593e-1 * c(6) - 0.1781799513347360596249022e-2 * c(5) - 0.3055555555555555555555556e-1 * c(9) 0.4698146218022683933926520e-1 * c(6) - 0.2966863787471237458744416e0 * c(8) + 0.5000000000000000000000000e-1 * c(10) + 0.1716355704146006481727960e0 * c(7) + 0.3069346152296258362380356e-2 * c(5) + 0.1750000000000000000000000e0 * c(9) -0.2308067892671916339568942e-1 * c(6) - 0.2986625053775149497180439e0 * c(7) - 0.3000000000000000000000000e0 * c(10) - 0.2500000000000000000000000e-1 * c(11) - 0.1047734860515050802561078e-2 * c(5) - 0.4272090808352508360837056e0 * c(8) - 0.4250000000000000000000000e0 * c(9) 0.5139370221149109977041877e-2 * c(6) + 0.1247723215009422001393184e0 * c(7) + 0.9505522702088127090209264e0 * c(8) + 0.9500000000000000000000000e0 * c(10) + 0.1250000000000000000000000e0 * c(11) + 0.5555555555555555555555556e-2 * c(12) + 0.9159362465153641826887659e-4 * c(5) + 0.5611111111111111111111111e0 * c(9); + ]; + + M(m-8:m,m-8:m)=[ + 0.5555555555555555555555556e-2 * c(m-11) + 0.1250000000000000000000000e0 * c(m-10) + 0.9500000000000000000000000e0 * c(m-9) + 0.9505522702088127090209264e0 * c(m-7) + 0.1247205076844361998744053e0 * c(m-6) + 0.5139370221149109977041877e-2 * c(m-5) + 0.5611111111111111111111111e0 * c(m-8) + 0.1434074411575366831819799e-3 * c(m-4) -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4); + -0.2500000000000000000000000e-1 * c(m-10) - 0.3000000000000000000000000e0 * c(m-9) - 0.2980649679116425253322056e0 * c(m-6) - 0.2308067892671916339568942e-1 * c(m-5) - 0.4250000000000000000000000e0 * c(m-8) - 0.4272090808352508360837056e0 * c(m-7) - 0.1645272326387475188399322e-2 * c(m-4) 0.5555555555555555555555556e-2 * c(m-10) + 0.1250000000000000000000000e0 * c(m-9) + 0.9500000000000000000000000e0 * c(m-8) + 0.9341601509609901526962449e0 * c(m-6) + 0.1183647529645898332481833e0 * c(m-5) + 0.1919432828897222527630486e-1 * c(m-4) + 0.5699474344521144554459336e0 * c(m-7) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6); + 0.5000000000000000000000000e-1 * c(m-9) - 0.2966863787471237458744416e0 * c(m-7) + 0.4698146218022683933926520e-1 * c(m-5) + 0.1750000000000000000000000e0 * c(m-8) + 0.1700291833903489463825077e0 * c(m-6) + 0.4675733176547960152668626e-2 * c(m-4) -0.2500000000000000000000000e-1 * c(m-9) - 0.3000000000000000000000000e0 * c(m-8) - 0.2997556885134827361576001e0 * c(m-5) - 0.5636663150858098975790317e-1 * c(m-4) - 0.4382544850115050165022338e0 * c(m-7) - 0.3806231949664312575822630e0 * c(m-6) 0.5555555555555555555555556e-2 * c(m-9) + 0.1250000000000000000000000e0 * c(m-8) + 0.9698817275172575247533506e0 * c(m-7) + 0.1011699483929608164601067e1 * c(m-5) + 0.1773466968705924819112984e0 * c(m-4) + 0.2239223735771599178951297e-3 * c(m-3) + 0.4325148359756313354830552e0 * c(m-6) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6); + 0.2279091916474916391629437e-1 * c(m-7) + 0.3097763128598982561225538e-1 * c(m-6) - 0.3055555555555555555555556e-1 * c(m-8) - 0.3068985997518740530511593e-1 * c(m-5) - 0.3634246031107139778989373e-2 * c(m-4) 0.5000000000000000000000000e-1 * c(m-8) - 0.3557251496099816106154206e0 * c(m-6) + 0.5490976528821799120017102e-1 * c(m-4) + 0.1838363233410033443348225e0 * c(m-7) + 0.2169790609807602750804271e0 * c(m-5) -0.2500000000000000000000000e-1 * c(m-8) - 0.3132544850115050165022338e0 * c(m-7) - 0.2322389872063761557916742e0 * c(m-4) - 0.8607044252686413302647675e-3 * c(m-3) - 0.2594851141920572702679681e0 * c(m-6) - 0.6691607091647929161078591e0 * c(m-5) 0.5555555555555555555555556e-2 * c(m-8) + 0.1338363233410033443348225e0 * c(m-7) + 0.7391887916719206077121040e0 * c(m-6) + 0.6490333320052011212240632e0 * c(m-4) + 0.3308343404200968256656458e-2 * c(m-3) + 0.2062575706647430620228133e-3 * c(m-2) + 0.3088241944378964404772302e-3 * c(m-1) + 0.4227226173449345042468960e-3 * c(m) + 0.1190362071861893051132274e1 * c(m-5) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5); + 0.5522702088127090209264064e-3 * c(m-7) - 0.3265435411305071914756373e-2 * c(m-6) + 0.7462059484530855073291365e-2 * c(m-5) - 0.4748894282038492179461399e-2 * c(m-4) 0.5528052133944605740009217e-1 * c(m-6) - 0.8631683980217122275970376e-1 * c(m-5) - 0.3276463639080639163926118e-1 * c(m-7) + 0.5268984374242044588776166e-1 * c(m-4) 0.5331362125287625412555844e-1 * c(m-7) + 0.1736073411355428563685818e0 * c(m-5) + 0.8671038084174692625075159e-2 * c(m-3) + 0.8084259844422177692569663e-1 * c(m-6) - 0.1664345989168155800449120e0 * c(m-4) -0.2720908083525083608370563e-1 * c(m-7) - 0.1931148612480615118957263e0 * c(m-6) - 0.3332941113251635390801278e-1 * c(m-3) + 0.2605582646183255957264249e-3 * c(m-2) + 0.1432116665752147607469646e-2 * c(m-1) + 0.1207544072304193806052558e-2 * c(m) - 0.1348436986667115543203552e1 * c(m-5) + 0.1687723507780044227927853e-1 * c(m-4) 0.6107825764368264576481962e-2 * c(m-7) + 0.1155752633643216628010304e0 * c(m-6) + 0.2096413329579026439044119e1 * c(m-5) + 0.3357721707576477199985656e0 * c(m-3) + 0.3291545083271862858501887e-3 * c(m-2) + 0.6641183499427826101618457e-2 * c(m-1) + 0.3449455095910233625229891e-2 * c(m) + 0.8270696421223286922584620e0 * c(m-4) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4); + 0.6272075574042975468177820e-3 * c(m-6) - 0.1200429618441003833696998e-1 * c(m-5) + 0.1137708862700574079015220e-1 * c(m-4) -0.5373770512016897565958305e-2 * c(m-6) + 0.1453421858063658498587377e0 * c(m-5) - 0.1399684152943489522927794e0 * c(m-4) -0.5013247356072127938999311e0 * c(m-5) + 0.5021853752328231128475915e0 * c(m-4) - 0.1197175073672143005877150e-1 * c(m-6) 0.3590669644811151307464697e-1 * c(m-6) - 0.5925443480724830632401754e0 * c(m-4) - 0.6897142765790609546343709e-2 * c(m-2) - 0.5363098747528542488971874e-2 * c(m-1) - 0.5205147429855955657625694e-2 * c(m) + 0.9977064356292750529201981e0 * c(m-5) -0.4995827370863505253765970e-1 * c(m-6) - 0.1263507837371824205693950e1 * c(m-5) - 0.8712928907711754187084757e-2 * c(m-2) - 0.2487040599390160764166412e-1 * c(m-1) - 0.1486895819265604128572498e-1 * c(m) - 0.1726565392121567634950213e1 * c(m-4) 0.2760393423824887721078848e-1 * c(m-6) + 0.1190550338687608873798462e1 * c(m-5) + 0.4253084328734353394994388e1 * c(m-4) + 0.2306367624634749229113646e0 * c(m-2) + 0.9313657638804699948929701e-1 * c(m-1) + 0.6409299775987186986730499e-1 * c(m) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6); + 0.9209089963443799485648361e-2 * c(m-5) - 0.3129629392354775191148163e-3 * c(m-6) - 0.8896127024208321966533544e-2 * c(m-4) -0.1059816030196818445908057e0 * c(m-5) + 0.1014880675788250237247178e0 * c(m-4) + 0.4493535440856820866087846e-2 * c(m-6) 0.3328335104489738933610597e0 * c(m-5) - 0.3179803804558436283847901e0 * c(m-4) - 0.5111353189352474549563559e-2 * c(m-3) - 0.9741776803777790426705996e-2 * c(m-6) 0.7272438906214475928744770e-1 * c(m-4) + 0.1964682777744275219350831e-1 * c(m-3) - 0.5952475275883259619711594e-2 * c(m-1) - 0.1635430866921887819487473e-2 * c(m) + 0.2921234010758621482958052e-1 * c(m-6) - 0.4659516693228870973898560e0 * c(m-5) 0.5402985338373433052255418e0 * c(m-5) - 0.1979290298620869974478871e0 * c(m-3) - 0.2760353365637712827793337e-1 * c(m-1) - 0.4671751091575462868310238e-2 * c(m) - 0.6952587985456154591014641e-1 * c(m-6) + 0.1571507277911208446562686e1 * c(m-4) -0.8755807343482262259774782e0 * c(m-5) - 0.3645285178085761821545207e1 * c(m-4) + 0.1033717994630886401730470e0 * c(m-1) + 0.2013769413884797246646959e-1 * c(m) + 0.4106783858513785463625543e-1 * c(m-6) 0.1070920689960817104203947e1 * c(m-5) + 0.3717418466925056542408153e1 * c(m-4) + 0.1166740554279680007487795e0 * c(m-3) + 0.1147318200715868527529827e0 * c(m-1) + 0.6327161147136873807796515e-2 * c(m) + 0.6235373239336055200426697e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6); + -0.3576545132696983143406173e-2 * c(m-5) + 0.4335019854436220306755673e-3 * c(m-6) + 0.3143043147253361112730605e-2 * c(m-4) 0.4093499466767054661591066e-1 * c(m-5) - 0.3471075437892810033585296e-1 * c(m-4) - 0.6224240288742446280057699e-2 * c(m-6) -0.1080774142196007991746827e0 * c(m-5) + 0.9941834083648937298100811e-1 * c(m-4) - 0.4834791406446907590553793e-2 * c(m-3) + 0.1349386478955833378422842e-1 * c(m-6) 0.5891947149681041048896399e-1 * c(m-4) + 0.1858378996391679448655070e-1 * c(m-3) + 0.7240905383565181316381731e-2 * c(m-2) + 0.2349927974590068869356781e-1 * c(m) - 0.4046360079256766884300687e-1 * c(m-6) + 0.1223513270418807666970488e0 * c(m-5) -0.1319358558853174530078498e0 * c(m-5) - 0.1872196143003808021730728e0 * c(m-3) + 0.9147192682075630179962131e-2 * c(m-2) + 0.6712774475803763988977355e-1 * c(m) + 0.9630407686703666967100804e-1 * c(m-6) - 0.6882132817757726722141421e0 * c(m-4) 0.3956598149904136332753521e0 * c(m-5) + 0.1630560443616104907615866e1 * c(m-4) - 0.2421320004064592721552708e0 * c(m-2) - 0.2893557395653431666593814e0 * c(m) - 0.5688529641249387985434413e-1 * c(m-6) -0.6639605248735044787146222e0 * c(m-5) - 0.1865625445986772763641423e1 * c(m-4) + 0.1103611313171476425250639e0 * c(m-3) - 0.9091410269992464604926176e-1 * c(m) - 0.8636954541126674177407762e-1 * c(m-6) 0.4681819359722749441073885e0 * c(m-5) + 0.1015239189167790053447110e1 * c(m-4) + 0.1043897828092562609502636e0 * c(m-3) + 0.2542001760457345743492403e0 * c(m-2) + 0.1306332157111667628555907e1 * c(m) + 0.1196351539550049336518187e0 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6); + 0.5593983696629863059347067e-3 * c(m-5) - 0.1446656414398166805849327e-3 * c(m-6) - 0.4147327282231696253497740e-3 * c(m-4) -0.6486184157331537899459796e-2 * c(m-5) + 0.4409068609809831485979484e-2 * c(m-4) + 0.2077115547521706413480312e-2 * c(m-6) 0.1319674981073749167009902e-1 * c(m-5) - 0.1060554802883657391328704e-1 * c(m-4) + 0.1911888563316170927411831e-2 * c(m-3) - 0.4503090345217088684223814e-2 * c(m-6) -0.2404661162020836566908542e-1 * c(m-4) - 0.7348845587775519698437916e-2 * c(m-3) - 0.8105784530576404277872603e-3 * c(m-2) + 0.9574633163221758060736592e-2 * c(m-1) - 0.1828896813877197352675410e-1 * c(m) + 0.1350326632905990039353503e-1 * c(m-6) - 0.1315967038382618382356495e-1 * c(m-5) 0.1241625568998496895352046e-1 * c(m-5) + 0.7403484645316174090533193e-1 * c(m-3) - 0.1023976547309387874453988e-2 * c(m-2) + 0.4440063948509876221050939e-1 * c(m-1) - 0.5224403464202056316702078e-1 * c(m) - 0.3213800979246298453953842e-1 * c(m-6) + 0.1178181682424363524005403e0 * c(m-4) -0.7684117160199014594442072e-1 * c(m-5) - 0.2928439026361256842196229e0 * c(m-4) + 0.2710530961648671297733465e-1 * c(m-2) - 0.1662748711097054895317080e0 * c(m-1) + 0.2251991532891353212689574e0 * c(m) + 0.1898341454096471754822498e-1 * c(m-6) 0.1582127073537215443965653e0 * c(m-5) + 0.3746489300753517635549495e0 * c(m-4) - 0.4364163147111892346990101e-1 * c(m-3) - 0.1845476106024151050283847e0 * c(m-1) + 0.7075642937243715046279337e-1 * c(m) + 0.2882271848190011329385407e-1 * c(m-6) -0.1195777325611201766551392e0 * c(m-5) - 0.2187310061229745694542609e0 * c(m-4) - 0.4128029838349298819825156e-1 * c(m-3) - 0.2845627370491611369031341e-1 * c(m-2) - 0.1016689339350338144430605e1 * c(m) - 0.3992391469197282238624438e-1 * c(m-6) 0.3167964748016105299646518e-1 * c(m-5) + 0.4976563420877041544013670e-1 * c(m-4) + 0.1632404042590951953384672e-1 * c(m-3) + 0.3185519088796429015220016e-2 * c(m-2) + 0.2968472090638000742888467e0 * c(m-1) + 0.7912667594695582093926295e0 * c(m) + 0.1332316557164627464149716e-1 * c(m-6); + ]; + + M(5,10)=M(10,5); + M(m-4,m-9)=M(m-9,m-4); + + M=M/h; + + D2 = HI*(-M - c(1)*e_l*d1_l' + c(m)*e_r*d1_r'); + end + D2 = @D2_fun; + + % Fourth derivative, 1th order accurate at first 8 boundary points (still + % yield 5th order convergence if stable: for example u_tt=-u_xxxx + stencil = [7/240, -2/5, 169/60, -122/15, 91/8, -122/15, 169/60, -2/5, 7/240]; + diags = -4:4; + M4 = stripeMatrix(stencil, diags, m); + + M4_U = [ + 0.1394226315049e13/0.367201486080e12 -0.1137054563243e13/0.114750464400e12 0.16614189027367e14/0.1836007430400e13 -0.1104821700277e13/0.306001238400e12 0.1355771086763e13/0.1836007430400e13 -0.27818686453e11/0.459001857600e12 -0.40671054239e11/0.1836007430400e13 0.5442887371e10/0.306001238400e12; + -0.1137054563243e13/0.114750464400e12 0.70616795535409e14/0.2570410402560e13 -0.173266854731041e15/0.6426026006400e13 0.28938615291031e14/0.2570410402560e13 -0.146167361863e12/0.71400288960e11 0.2793470836571e13/0.12852052012800e14 0.6219558097e10/0.428401733760e12 -0.7313844559e10/0.166909766400e12; + 0.16614189027367e14/0.1836007430400e13 -0.173266854731041e15/0.6426026006400e13 0.378613061504779e15/0.12852052012800e14 -0.9117069604217e13/0.642602600640e12 0.632177582849e12/0.233673672960e12 -0.1057776382577e13/0.6426026006400e13 0.443019868399e12/0.4284017337600e13 -0.3707981e7/0.2318191200e10; + -0.1104821700277e13/0.306001238400e12 0.28938615291031e14/0.2570410402560e13 -0.9117069604217e13/0.642602600640e12 0.5029150721885e13/0.514082080512e12 -0.5209119714341e13/0.1285205201280e13 0.12235427457469e14/0.12852052012800e14 -0.13731270505e11/0.64260260064e11 0.2933596129e10/0.40800165120e11; + 0.1355771086763e13/0.1836007430400e13 -0.146167361863e12/0.71400288960e11 0.632177582849e12/0.233673672960e12 -0.5209119714341e13/0.1285205201280e13 0.14871726798559e14/0.2570410402560e13 -0.7504337615347e13/0.1606506501600e13 0.310830296467e12/0.171360693504e12 -0.55284274391e11/0.183600743040e12; + -0.27818686453e11/0.459001857600e12 0.2793470836571e13/0.12852052012800e14 -0.1057776382577e13/0.6426026006400e13 0.12235427457469e14/0.12852052012800e14 -0.7504337615347e13/0.1606506501600e13 0.106318657014853e15/0.12852052012800e14 -0.14432772918527e14/0.2142008668800e13 0.58102695589e11/0.22666758400e11; + -0.40671054239e11/0.1836007430400e13 0.6219558097e10/0.428401733760e12 0.443019868399e12/0.4284017337600e13 -0.13731270505e11/0.64260260064e11 0.310830296467e12/0.171360693504e12 -0.14432772918527e14/0.2142008668800e13 0.27102479467823e14/0.2570410402560e13 -0.1216032192203e13/0.153000619200e12; + 0.5442887371e10/0.306001238400e12 -0.7313844559e10/0.166909766400e12 -0.3707981e7/0.2318191200e10 0.2933596129e10/0.40800165120e11 -0.55284274391e11/0.183600743040e12 0.58102695589e11/0.22666758400e11 -0.1216032192203e13/0.153000619200e12 0.20799922829107e14/0.1836007430400e13; + ]; + + M4(1:8,1:8) = M4_U; + M4(m-7:m,m-7:m) = rot90( M4_U ,2 ); + M4 = M4/h^3; + + + + D4=HI*(M4 - e_l*d3_l'+e_r*d3_r' + d1_l*d2_l'-d1_r*d2_r'); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D1Nonequidistant.m --- a/+sbp/D1Nonequidistant.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/D1Nonequidistant.m Tue Feb 07 16:09:02 2017 +0100 @@ -81,6 +81,11 @@ obj.Q = obj.H*obj.D1 - obj.e_r*obj.e_r' + obj.e_l*obj.e_l'; obj.borrowing = []; + + end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; end end end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D2Standard.m --- a/+sbp/D2Standard.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/D2Standard.m Tue Feb 07 16:09:02 2017 +0100 @@ -64,10 +64,10 @@ obj.m = m; end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; + end + end end - - - - -
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D2Variable.m --- a/+sbp/D2Variable.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/D2Variable.m Tue Feb 07 16:09:02 2017 +0100 @@ -18,7 +18,7 @@ methods function obj = D2Variable(m,lim,order) - + x_l = lim{1}; x_r = lim{2}; L = x_r-x_l; @@ -30,12 +30,12 @@ [obj.H, obj.HI, obj.D1, obj.D2, obj.e_l,... obj.e_r, obj.d1_l, obj.d1_r] = ... sbp.implementations.d2_variable_4(m,obj.h); - obj.borrowing.M.S = 0.2505765857; + obj.borrowing.M.d1 = 0.2505765857; case 2 [obj.H, obj.HI, obj.D1, obj.D2, obj.e_l,... obj.e_r, obj.d1_l, obj.d1_r] = ... sbp.implementations.d2_variable_2(m,obj.h); - obj.borrowing.M.S = 0.3636363636; + obj.borrowing.M.d1 = 0.3636363636; % Borrowing const taken from Virta 2014 otherwise @@ -44,10 +44,13 @@ obj.m = m; obj.M = []; - + end + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; end end + end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D4CompatibleVariable.m --- a/+sbp/D4CompatibleVariable.m Tue Feb 07 15:47:51 2017 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,74 +0,0 @@ -classdef D4CompatibleVariable < sbp.OpSet - properties - D1 % SBP operator approximating first derivative - H % Norm matrix - HI % H^-1 - Q % Skew-symmetric matrix - e_l % Left boundary operator - e_r % Right boundary operator - D2 % SBP operator for second derivative - M % Norm matrix, second derivative - d1_l % Left boundary first derivative - d1_r % Right boundary first derivative - D3 % SBP operator for third derivative - Q3 % Skew-symmetric matrix in third derivative - d2_l % Left boundary second derivative - d2_r % Right boundary second derivative - D4 % SBP operator for fourth derivative - M4 % Norm matrix, fourth derivative - d3_l % Left boundary third derivative - d3_r % Right boundary third derivative - m % Number of grid points. - h % Step size - x % grid - borrowing % Struct with borrowing limits for different norm matrices - end - - - - methods - function obj = D4CompatibleVariable(m,lim,order) - - x_l = lim{1}; - x_r = lim{2}; - L = x_r-x_l; - obj.h = L/(m-1); - obj.x = linspace(x_l,x_r,m)'; - - if order == 2 - [obj.H, obj.HI, ~, obj.D2, ~, obj.D4, obj.e_l, obj.e_r,... - obj.M4, ~, obj.d2_l, obj.d2_r, obj.d3_l,... - obj.d3_r, obj.d1_l, obj.d1_r] =... - sbp.implementations.d4_compatible_halfvariable_2(m,obj.h); - obj.borrowing.N.S2 = 1.2500; - obj.borrowing.N.S3 = 0.4000; - elseif order == 4 - [obj.H, obj.HI, obj.D2, obj.D4, obj.e_l, obj.e_r, obj.M4,... - obj.d2_l, obj.d2_r, obj.d3_l, obj.d3_r, obj.d1_l,... - obj.d1_r] =... - sbp.implementations.d4_compatible_halfvariable_4(m,obj.h); - obj.borrowing.N.S2 = 0.5055; - obj.borrowing.N.S3 = 0.9290; - elseif order == 6 - [obj.H, obj.HI, obj.D2, obj.D4, obj.e_l, obj.e_r, obj.M4,... - obj.d2_l, obj.d2_r, obj.d3_l, obj.d3_r, obj.d1_l,... - obj.d1_r] =... - sbp.implementations.d4_compatible_halfvariable_6(m,obj.h); - obj.borrowing.N.S2 = 0.3259; - obj.borrowing.N.S3 = 0.1580; - else - error('Invalid operator order.'); - end - - obj.m = m; - - obj.D1 = []; - obj.D3 = []; - - - end - end - - - -end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D4Lonely.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/D4Lonely.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,124 @@ +classdef D4Lonely < sbp.OpSet + properties + m % Number of grid points. + h % Step size + x % grid + H % Norm matrix + HI % H^-1 + D4 % SBP operator for fourth derivative + M4 % Norm matrix, fourth derivative + e_l, e_r % Left and right boundary operator + d1_l, d1_r % Left and right boundary first derivative + d2_l, d2_r % Left and right boundary second derivative + d3_l, d3_r % Left and right boundary third derivative + borrowing % Struct with borrowing limits for different norm matrices + opt + order + end + + methods + function obj = D4Lonely(m, lim, order, opt) + default_arg('opt', '') + + obj.opt = opt; + obj.order = order; + + x_l = lim{1}; + x_r = lim{2}; + L = x_r-x_l; + obj.h = L/(m-1); + obj.x = linspace(x_l, x_r,m)'; + + if order == 2 + switch opt + case '' + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_2(m, obj.h); + obj.borrowing.N.S2 = 1.2500; + obj.borrowing.N.S3 = 0.4000; + otherwise + error('Invalid operator option.'); + end + + elseif order == 4 + switch opt + case 'min_boundary_points' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_4_min_boundary_points(m, obj.h); + obj.borrowing.N.S2 = 0.6244; + obj.borrowing.N.S3 = 1.3961; + case '' + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_4(m, obj.h); + obj.borrowing.N.S2 = 0.5055; + obj.borrowing.N.S3 = 0.9290; + otherwise + error('Invalid operator option.'); + end + + elseif order == 6 + switch opt + case '2' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_6_2(m, obj.h); + obj.borrowing.N.S2 = 0.2931; + obj.borrowing.N.S3 = 0.0807; + case '3' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_6_3(m, obj.h); + obj.borrowing.N.S2 = 0.2842; + obj.borrowing.N.S3 = 0.0709; + case 'min_boundary_points' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_6_min_boundary_points(m, obj.h); + obj.borrowing.N.S2 = 0.3569; + obj.borrowing.N.S3 = 0.1908; + case '' + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_6(m, obj.h); + obj.borrowing.N.S2 = 0.3259; + obj.borrowing.N.S3 = 0.1580; + otherwise + error('Invalid operator option.'); + end + + elseif order == 8 + switch opt + case 'min_boundary_points' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_8_min_boundary_points(m, obj.h); + obj.borrowing.N.S2 = 0.2804; + obj.borrowing.N.S3 = 0.0740; + case '' + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_lonely_8_higher_boundary_order(m, obj.h); + obj.borrowing.N.S2 = 0.2475; + obj.borrowing.N.S3 = 0.0401; + otherwise + error('Invalid operator option.'); + end + else + error('Invalid operator order.'); + end + + obj.m = m; + + obj.H = H; + obj.HI = HI; + obj.D4 = D4; + obj.M4 = M4; + obj.e_l = e_l; + obj.e_r = e_r; + obj.d1_l = d1_l; + obj.d1_r = d1_r; + obj.d2_l = d2_l; + obj.d2_r = d2_r; + obj.d3_l = d3_l; + obj.d3_r = d3_r; + end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order) '_' obj.opt]; + end + end +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/D4Variable.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/D4Variable.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,75 @@ +classdef D4Variable < sbp.OpSet + properties + m % Number of grid points. + h % Step size + x % grid + H % Norm matrix + HI % H^-1 + D1 % SBP operator approximating first derivative + D2 % SBP operator for second derivative + D4 % SBP operator for fourth derivative + Q % Skew-symmetric matrix + M % Norm matrix, second derivative + M4 % Norm matrix, fourth derivative + e_l, e_r % Left and right boundary operator + d1_l, d1_r % Left and right boundary first derivative + d2_l, d2_r % Left and right boundary second derivative + d3_l, d3_r % Left and right boundary third derivative + borrowing % Struct with borrowing limits for different norm matrices + order + end + + methods + function obj = D4Variable(m, lim, order) + x_l = lim{1}; + x_r = lim{2}; + L = x_r-x_l; + obj.h = L/(m-1); + obj.x = linspace(x_l, x_r,m)'; + + if order == 2 + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_2(m, obj.h); + obj.borrowing.M.d1 = 0.4000; + obj.borrowing.N.S2 = 1.2500; + obj.borrowing.N.S3 = 0.4000; + elseif order == 4 + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_4(m, obj.h); + obj.borrowing.M.d1 = 0.2508; + obj.borrowing.N.S2 = 0.5055; + obj.borrowing.N.S3 = 0.9290; + elseif order == 6 + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = ... + sbp.implementations.d4_variable_6(m, obj.h); + obj.borrowing.M.d1 = 0.1878; + obj.borrowing.N.S2 = 0.3259; + obj.borrowing.N.S3 = 0.1580; + else + error('Invalid operator order.'); + end + + obj.m = m; + obj.order = order; + + obj.H = H; + obj.HI = HI; + obj.D1 = D1; + obj.D2 = D2; + obj.D4 = D4; + obj.M4 = M4; + obj.e_l = e_l; + obj.e_r = e_r; + obj.d1_l = d1_l; + obj.d1_r = d1_r; + obj.d2_l = d2_l; + obj.d2_r = d2_r; + obj.d3_l = d3_l; + obj.d3_r = d3_r; + end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; + end + end +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +sbp/OpSet.m --- a/+sbp/OpSet.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+sbp/OpSet.m Tue Feb 07 16:09:02 2017 +0100 @@ -6,4 +6,9 @@ x % Grid end + methods (Abstract) + % Returns a string representation of the type of operator. + str = string(obj) + end + end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +scheme/Beam.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+scheme/Beam.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,222 @@ +classdef Beam < scheme.Scheme + properties + order % Order accuracy for the approximation + grid + + D % non-stabalized scheme operator + alpha + + h + H % Discrete norm + Hi + + e_l, e_r + d1_l, d1_r + d2_l, d2_r + d3_l, d3_r + gamm + delt + alphaII + alphaIII + + opt + end + + methods + function obj = Beam(grid, order, alpha, opsGen, opt) + default_arg('alpha', -1); + + % default_arg('opsGen', @sbp.D4); + default_arg('opsGen', @sbp.D4Variable); % Supposed to be better + + opt_default.interface_l.tuning = 1.1; + opt_default.interface_l.tau = []; + opt_default.interface_l.sig = []; + opt_default.interface_r.tuning = 1.1; + opt_default.interface_r.tau = []; + opt_default.interface_r.sig = []; + default_struct('opt', opt_default); + + if ~isa(grid, 'grid.Cartesian') || grid.D() ~= 1 + error('Grid must be 1d cartesian'); + end + + obj.grid = grid; + obj.order = order; + obj.alpha = alpha; + + m = grid.m; + h = grid.scaling(); + + x_lim = {grid.x{1}(1), grid.x{1}(end)}; + ops = opsGen(m, x_lim, order); + + D4 = ops.D4; + obj.H = ops.H; + obj.Hi = ops.HI; + obj.e_l = ops.e_l; + obj.e_r = ops.e_r; + obj.d1_l = ops.d1_l; + obj.d1_r = ops.d1_r; + obj.d2_l = ops.d2_l; + obj.d2_r = ops.d2_r; + obj.d3_l = ops.d3_l; + obj.d3_r = ops.d3_r; + + obj.D = alpha*D4; + + alphaII = ops.borrowing.N.S2/2; + alphaIII = ops.borrowing.N.S3/2; + + obj.gamm = h*alphaII; + obj.delt = h^3*alphaIII; + obj.alphaII = alphaII; + obj.alphaIII = alphaIII; + obj.h = h; + obj.opt = opt; + end + + + % Closure functions return the opertors applied to the own doamin to close the boundary + % Penalty functions return the opertors to force the solution. In the case of an interface it returns the operator applied to the other doamin. + % boundary is a string specifying the boundary e.g. 'l','r' or 'e','w','n','s'. + % type is a string specifying the type of boundary condition if there are several. + % neighbour_scheme is an instance of Scheme that should be interfaced to. + % neighbour_boundary is a string specifying which boundary to interface to. + function [closure, penalty] = boundary_condition(obj,boundary,type) + default_arg('type','dn'); + + [e, d1, d2, d3, s] = obj.get_boundary_ops(boundary); + gamm = obj.gamm; + delt = obj.delt; + + switch type + case {'dn', 'clamped'} % Dirichlet-neumann boundary condition + alpha = obj.alpha; + + % tau1 < -alpha^2/gamma + % tuning = 2; + tuning = 1.1; + + tau1 = tuning * alpha/delt; + tau4 = s*alpha; + + sig2 = tuning * alpha/gamm; + sig3 = -s*alpha; + + tau = tau1*e+tau4*d3; + sig = sig2*d1+sig3*d2; + + closure = obj.Hi*(tau*e' + sig*d1'); + + penalty{1} = -obj.Hi*tau; + penalty{2} = -obj.Hi*sig; + + + case {'free'} + a = obj.alpha; + + tau = s*a*d1; + sig = -s*a*e; + + closure = obj.Hi*(tau*d2' + sig*d3'); + penalty{1} = -obj.Hi*tau; + penalty{1} = -obj.Hi*sig; + + + otherwise % Unknown, boundary condition + error('No such boundary condition: type = %s',type); + end + end + + function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + % u denotes the solution in the own domain + % v denotes the solution in the neighbour domain + [e_u,d1_u,d2_u,d3_u,s_u] = obj.get_boundary_ops(boundary); + [e_v,d1_v,d2_v,d3_v,s_v] = neighbour_scheme.get_boundary_ops(neighbour_boundary); + + + alpha_u = obj.alpha; + alpha_v = neighbour_scheme.alpha; + + + switch boundary + case 'l' + interface_opt = obj.opt.interface_l; + case 'r' + interface_opt = obj.opt.interface_r; + end + + + if isempty(interface_opt.tau) && isempty(interface_opt.sig) + gamm_u = obj.gamm; + delt_u = obj.delt; + + gamm_v = neighbour_scheme.gamm; + delt_v = neighbour_scheme.delt; + + tuning = interface_opt.tuning; + + tau1 = ((alpha_u/2)/delt_u + (alpha_v/2)/delt_v)/2*tuning; + sig2 = ((alpha_u/2)/gamm_u + (alpha_v/2)/gamm_v)/2*tuning; + else + h_u = obj.h; + h_v = neighbour_scheme.h; + + switch neighbour_boundary + case 'l' + neighbour_interface_opt = neighbour_scheme.opt.interface_l; + case 'r' + neighbour_interface_opt = neighbour_scheme.opt.interface_r; + end + + tau_u = interface_opt.tau; + sig_u = interface_opt.sig; + tau_v = neighbour_interface_opt.tau; + sig_v = neighbour_interface_opt.sig; + + tau1 = tau_u/h_u^3 + tau_v/h_v^3; + sig2 = sig_u/h_u + sig_v/h_v; + end + + tau4 = s_u*alpha_u/2; + sig3 = -s_u*alpha_u/2; + phi2 = s_u*1/2; + psi1 = -s_u*1/2; + + tau = tau1*e_u + tau4*d3_u; + sig = sig2*d1_u + sig3*d2_u ; + phi = phi2*d1_u ; + psi = psi1*e_u ; + + closure = obj.Hi*(tau*e_u' + sig*d1_u' + phi*alpha_u*d2_u' + psi*alpha_u*d3_u'); + penalty = -obj.Hi*(tau*e_v' + sig*d1_v' + phi*alpha_v*d2_v' + psi*alpha_v*d3_v'); + end + + % Returns the boundary ops and sign for the boundary specified by the string boundary. + % The right boundary is considered the positive boundary + function [e, d1, d2, d3, s] = get_boundary_ops(obj,boundary) + switch boundary + case 'l' + e = obj.e_l; + d1 = obj.d1_l; + d2 = obj.d2_l; + d3 = obj.d3_l; + s = -1; + case 'r' + e = obj.e_r; + d1 = obj.d1_r; + d2 = obj.d2_r; + d3 = obj.d3_r; + s = 1; + otherwise + error('No such boundary: boundary = %s',boundary); + end + end + + function N = size(obj) + N = obj.grid.N; + end + + end +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +scheme/Beam2d.m --- a/+scheme/Beam2d.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+scheme/Beam2d.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,10 +1,6 @@ classdef Beam2d < scheme.Scheme properties - m % Number of points in each direction, possibly a vector - N % Number of points total - h % Grid spacing - u,v % Grid - x,y % Values of x and y for each grid point + grid order % Order accuracy for the approximation D % non-stabalized scheme operator @@ -27,21 +23,23 @@ methods function obj = Beam2d(m,lim,order,alpha,opsGen) + default_arg('alpha',1); default_arg('opsGen',@sbp.Higher); - default_arg('a',1); - if length(m) == 1 - m = [m m]; + if ~isa(grid, 'grid.Cartesian') || grid.D() ~= 2 + error('Grid must be 2d cartesian'); end - m_x = m(1); - m_y = m(2); + obj.grid = grid; + obj.alpha = alpha; + obj.order = order; - xlim = lim{1}; - ylim = lim{2}; + m_x = grid.m(1); + m_y = grid.m(2); - [x, h_x] = util.get_grid(xlim{:},m_x); - [y, h_y] = util.get_grid(ylim{:},m_y); + h = grid.scaling(); + h_x = h(1); + h_y = h(2); ops_x = opsGen(m_x,h_x,order); ops_y = opsGen(m_y,h_y,order); @@ -49,9 +47,6 @@ I_x = speye(m_x); I_y = speye(m_y); - - - D4_x = sparse(ops_x.derivatives.D4); H_x = sparse(ops_x.norms.H); Hi_x = sparse(ops_x.norms.HI); @@ -105,16 +100,7 @@ obj.d3_s = kr(I_x,d3_l_y); obj.d3_n = kr(I_x,d3_r_y); - obj.m = m; - obj.h = [h_x h_y]; - obj.order = order; - - obj.alpha = alpha; obj.D = alpha*D4; - obj.u = x; - obj.v = y; - obj.x = kr(x,ones(m_y,1)); - obj.y = kr(ones(m_x,1),y); obj.gamm_x = h_x*ops_x.borrowing.N.S2/2; obj.delt_x = h_x^3*ops_x.borrowing.N.S3/2;
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +scheme/Scheme.m --- a/+scheme/Scheme.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+scheme/Scheme.m Tue Feb 07 16:09:02 2017 +0100 @@ -20,13 +20,12 @@ % 'l','r' or 'e','w','n','s'. % type is a string specifying the type of % boundary condition if there are several. - % data is a function returning the data that - % should be applied at the boundary. % neighbour_scheme is an instance of Scheme that should be % interfaced to. % neighbour_boundary is a string specifying which boundary to % interface to. - [closure, penalty] = boundary_condition(obj,boundary,type,data) + % penalty may be a cell array if there are several penalties with different weights + [closure, penalty] = boundary_condition(obj,boundary,type) [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) % Returns the number of degrees of freedom. @@ -42,4 +41,4 @@ [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u); end end -end \ No newline at end of file +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +scheme/Wave2dCurve.m --- a/+scheme/Wave2dCurve.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+scheme/Wave2dCurve.m Tue Feb 07 16:09:02 2017 +0100 @@ -2,9 +2,9 @@ properties m % Number of points in each direction, possibly a vector h % Grid spacing - u,v % Grid - x,y % Values of x and y for each grid point - X,Y % Grid point locations as matrices + + grid + order % Order accuracy for the approximation D % non-stabalized scheme operator @@ -26,24 +26,28 @@ du_n, dv_n gamm_u, gamm_v lambda + + x_u + x_v + y_u + y_v end methods - function obj = Wave2dCurve(m,ti,order,c,opSet) + function obj = Wave2dCurve(g ,order, c, opSet) default_arg('opSet',@sbp.D2Variable); default_arg('c', 1); - if length(m) == 1 - m = [m m]; - end + assert(isa(g, 'grid.Curvilinear')) + m = g.size(); m_u = m(1); m_v = m(2); - m_tot = m_u*m_v; + m_tot = g.N(); - [u, h_u] = util.get_grid(0, 1, m_u); - [v, h_v] = util.get_grid(0, 1, m_v); - + h = g.scaling(); + h_u = h(1); + h_v = h(2); % Operators ops_u = opSet(m_u, {0, 1}, order); @@ -52,66 +56,64 @@ I_u = speye(m_u); I_v = speye(m_v); - D1_u = sparse(ops_u.D1); + D1_u = ops_u.D1; D2_u = ops_u.D2; - H_u = sparse(ops_u.H); - Hi_u = sparse(ops_u.HI); - % M_u = sparse(ops_u.M); - e_l_u = sparse(ops_u.e_l); - e_r_u = sparse(ops_u.e_r); - d1_l_u = sparse(ops_u.d1_l); - d1_r_u = sparse(ops_u.d1_r); + H_u = ops_u.H; + Hi_u = ops_u.HI; + e_l_u = ops_u.e_l; + e_r_u = ops_u.e_r; + d1_l_u = ops_u.d1_l; + d1_r_u = ops_u.d1_r; - D1_v = sparse(ops_v.D1); + D1_v = ops_v.D1; D2_v = ops_v.D2; - H_v = sparse(ops_v.H); - Hi_v = sparse(ops_v.HI); - % M_v = sparse(ops_v.M); - e_l_v = sparse(ops_v.e_l); - e_r_v = sparse(ops_v.e_r); - d1_l_v = sparse(ops_v.d1_l); - d1_r_v = sparse(ops_v.d1_r); + H_v = ops_v.H; + Hi_v = ops_v.HI; + e_l_v = ops_v.e_l; + e_r_v = ops_v.e_r; + d1_l_v = ops_v.d1_l; + d1_r_v = ops_v.d1_r; + + Du = kr(D1_u,I_v); + Dv = kr(I_u,D1_v); % Metric derivatives - [X,Y] = ti.map(u,v); + coords = g.points(); + x = coords(:,1); + y = coords(:,2); - [x_u,x_v] = gridDerivatives(X,D1_u,D1_v); - [y_u,y_v] = gridDerivatives(Y,D1_u,D1_v); - - + x_u = Du*x; + x_v = Dv*x; + y_u = Du*y; + y_v = Dv*y; J = x_u.*y_v - x_v.*y_u; - a11 = 1./J .* (x_v.^2 + y_v.^2); %% GÖR SOM MATRISER + a11 = 1./J .* (x_v.^2 + y_v.^2); a12 = -1./J .* (x_u.*x_v + y_u.*y_v); a22 = 1./J .* (x_u.^2 + y_u.^2); lambda = 1/2 * (a11 + a22 - sqrt((a11-a22).^2 + 4*a12.^2)); - dof_order = reshape(1:m_u*m_v,m_v,m_u); + % Assemble full operators + L_12 = spdiags(a12, 0, m_tot, m_tot); + Duv = Du*L_12*Dv; + Dvu = Dv*L_12*Du; Duu = sparse(m_tot); Dvv = sparse(m_tot); + ind = grid.funcToMatrix(g, 1:m_tot); for i = 1:m_v - D = D2_u(a11(i,:)); - p = dof_order(i,:); + D = D2_u(a11(ind(:,i))); + p = ind(:,i); Duu(p,p) = D; end for i = 1:m_u - D = D2_v(a22(:,i)); - p = dof_order(:,i); + D = D2_v(a22(ind(i,:))); + p = ind(i,:); Dvv(p,p) = D; end - L_12 = spdiags(a12(:),0,m_tot,m_tot); - Du = kr(D1_u,I_v); - Dv = kr(I_u,D1_v); - - Duv = Du*L_12*Dv; - Dvu = Dv*L_12*Du; - - - obj.H = kr(H_u,H_v); obj.Hi = kr(Hi_u,Hi_v); obj.Hu = kr(H_u,I_v); @@ -119,7 +121,6 @@ obj.Hiu = kr(Hi_u,I_v); obj.Hiv = kr(I_u,Hi_v); - % obj.M = kr(M_u,H_v)+kr(H_u,M_v); obj.e_w = kr(e_l_u,I_v); obj.e_e = kr(e_r_u,I_v); obj.e_s = kr(I_u,e_l_v); @@ -133,28 +134,27 @@ obj.du_n = (obj.e_n'*Du)'; obj.dv_n = kr(I_u,d1_r_v); + obj.x_u = x_u; + obj.x_v = x_v; + obj.y_u = y_u; + obj.y_v = y_v; + obj.m = m; obj.h = [h_u h_v]; obj.order = order; - + obj.grid = g; obj.c = c; - obj.J = spdiags(J(:),0,m_tot,m_tot); - obj.Ji = spdiags(1./J(:),0,m_tot,m_tot); + obj.J = spdiags(J, 0, m_tot, m_tot); + obj.Ji = spdiags(1./J, 0, m_tot, m_tot); obj.a11 = a11; obj.a12 = a12; obj.a22 = a22; obj.D = obj.Ji*c^2*(Duu + Duv + Dvu + Dvv); - obj.u = u; - obj.v = v; - obj.X = X; - obj.Y = Y; - obj.x = X(:); - obj.y = Y(:); obj.lambda = lambda; - obj.gamm_u = h_u*ops_u.borrowing.M.S; - obj.gamm_v = h_v*ops_v.borrowing.M.S; + obj.gamm_u = h_u*ops_u.borrowing.M.d1; + obj.gamm_v = h_v*ops_v.borrowing.M.d1; end @@ -165,12 +165,11 @@ % data is a function returning the data that should be applied at the boundary. % neighbour_scheme is an instance of Scheme that should be interfaced to. % neighbour_boundary is a string specifying which boundary to interface to. - function [closure, penalty] = boundary_condition(obj,boundary,type,data) + function [closure, penalty] = boundary_condition(obj, boundary, type, parameter) default_arg('type','neumann'); - default_arg('data',0); + default_arg('parameter', []); - [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv] = obj.get_boundary_ops(boundary); - + [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv , ~, ~, ~, scale_factor] = obj.get_boundary_ops(boundary); switch type % Dirichlet boundary condition case {'D','d','dirichlet'} @@ -190,28 +189,19 @@ b2 = gamm*u.lambda./u.a22.^2; tau = -1./b1 - 1./b2; - tau = tuning * spdiag(tau(:)); - sig1 = 1/2; + tau = tuning * spdiag(tau); + sig1 = 1; penalty_parameter_1 = halfnorm_inv_n*(tau + sig1*halfnorm_inv_t*F*e'*halfnorm_t)*e; closure = obj.Ji*obj.c^2 * penalty_parameter_1*e'; - pp = -obj.Ji*obj.c^2 * penalty_parameter_1; - switch class(data) - case 'double' - penalty = pp*data; - case 'function_handle' - penalty = @(t)pp*data(t); - otherwise - error('Weird data argument!') - end + penalty = -obj.Ji*obj.c^2 * penalty_parameter_1; % Neumann boundary condition case {'N','n','neumann'} c = obj.c; - a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n)); a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t)); d = (a_n * d_n' + a_t*d_t')'; @@ -221,16 +211,23 @@ tau = c.^2 * obj.Ji*(tau1*e + tau2*d); closure = halfnorm_inv*tau*d'; + penalty = -halfnorm_inv*tau; - pp = halfnorm_inv*tau; - switch class(data) - case 'double' - penalty = pp*data; - case 'function_handle' - penalty = @(t)pp*data(t); - otherwise - error('Weird data argument!') - end + % Characteristic boundary condition + case {'characteristic', 'char', 'c'} + default_arg('parameter', 1); + beta = parameter; + c = obj.c; + + a_n = spdiags(coeff_n,0,length(coeff_n),length(coeff_n)); + a_t = spdiags(coeff_t,0,length(coeff_t),length(coeff_t)); + d = s*(a_n * d_n' + a_t*d_t')'; % outward facing normal derivative + + tau = -c.^2 * 1/beta*obj.Ji*e; + + closure{1} = halfnorm_inv*tau/c*spdiag(scale_factor)*e'; + closure{2} = halfnorm_inv*tau*beta*d'; + penalty = -halfnorm_inv*tau; % Unknown, boundary condition otherwise @@ -263,7 +260,7 @@ b2_v = gamm_v*v.lambda(I_v)./v.a22(I_v).^2; tau = -1./(4*b1_u) -1./(4*b1_v) -1./(4*b2_u) -1./(4*b2_v); - tau = tuning * spdiag(tau(:)); + tau = tuning * spdiag(tau); sig1 = 1/2; sig2 = -1/2; @@ -279,10 +276,12 @@ % The right boundary is considered the positive boundary % % I -- the indecies of the boundary points in the grid matrix - function [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t, I] = get_boundary_ops(obj,boundary) + function [e, d_n, d_t, coeff_n, coeff_t, s, gamm, halfnorm_inv_n, halfnorm_inv_t, halfnorm_t, I, scale_factor] = get_boundary_ops(obj, boundary) - gridMatrix = zeros(obj.m(2),obj.m(1)); - gridMatrix(:) = 1:numel(gridMatrix); + % gridMatrix = zeros(obj.m(2),obj.m(1)); + % gridMatrix(:) = 1:numel(gridMatrix); + + ind = grid.funcToMatrix(obj.grid, 1:prod(obj.m)); switch boundary case 'w' @@ -291,36 +290,40 @@ d_t = obj.dv_w; s = -1; - I = gridMatrix(:,1); + I = ind(1,:); coeff_n = obj.a11(I); coeff_t = obj.a12(I); + scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2); case 'e' e = obj.e_e; d_n = obj.du_e; d_t = obj.dv_e; s = 1; - I = gridMatrix(:,end); + I = ind(end,:); coeff_n = obj.a11(I); coeff_t = obj.a12(I); + scale_factor = sqrt(obj.x_v(I).^2 + obj.y_v(I).^2); case 's' e = obj.e_s; d_n = obj.dv_s; d_t = obj.du_s; s = -1; - I = gridMatrix(1,:)'; + I = ind(:,1)'; coeff_n = obj.a22(I); coeff_t = obj.a12(I); + scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2); case 'n' e = obj.e_n; d_n = obj.dv_n; d_t = obj.du_n; s = 1; - I = gridMatrix(end,:)'; + I = ind(:,end)'; coeff_n = obj.a22(I); coeff_t = obj.a12(I); + scale_factor = sqrt(obj.x_u(I).^2 + obj.y_u(I).^2); otherwise error('No such boundary: boundary = %s',boundary); end @@ -343,15 +346,6 @@ N = prod(obj.m); end - end - methods(Static) - % Calculates the matrcis need for the inteface coupling between boundary bound_u of scheme schm_u - % and bound_v of scheme schm_v. - % [uu, uv, vv, vu] = inteface_couplong(A,'r',B,'l') - function [uu, uv, vv, vu] = interface_coupling(schm_u,bound_u,schm_v,bound_v) - [uu,uv] = schm_u.interface(bound_u,schm_v,bound_v); - [vv,vu] = schm_v.interface(bound_v,schm_u,bound_u); - end end end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/Cdiff.m --- a/+time/Cdiff.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+time/Cdiff.m Tue Feb 07 16:09:02 2017 +0100 @@ -12,22 +12,20 @@ methods + % Solves u_tt = Du + Eu_t + S + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. + % Cdiff(D, E, S, k, t0, n0, v, v_prev) function obj = Cdiff(D, E, S, k, t0, n0, v, v_prev) - % Cdiff(D, E, S, k, t0, n0, v, v_prev) - m = size(D,1); + m = length(v); default_arg('E',sparse(m,m)); default_arg('S',sparse(m,1)); - if ~(issparse(D) && issparse(E) && issparse(S)) - warning('One of the matrices D, E, S is not sparse!') - print_issparse(D) - print_issparse(E) - print_issparse(S) - end - obj.D = D; obj.E = E; obj.S = S; + + obj.k = k; obj.t = t0; obj.n = n0;
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/CdiffImplicit.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+time/CdiffImplicit.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,150 @@ +classdef CdiffImplicit < time.Timestepper + properties + A, B, C, G + AA, BB, CC + k + t + v, v_prev + n + + % LU factorization + L,U,p,q + end + + methods + % Solves + % A*u_tt + B*u + C*v_t = G(t) + % u(t0) = f1 + % u_t(t0) = f2 + % starting at time t0 with timestep k + function obj = CdiffImplicit(A, B, C, G, f1, f2, k, t0) + default_arg('A', []); + default_arg('C', []); + default_arg('G', []); + default_arg('f1', 0); + default_arg('f2', 0); + default_arg('t0', 0); + + m = size(B,1); + + if isempty(A) + A = speye(m); + end + + if isempty(C) + C = sparse(m,m); + end + + if isempty(G) + G = @(t) sparse(m,1); + end + + if isempty(f1) + f1 = sparse(m,1); + end + + if isempty(f2) + f2 = sparse(m,1); + end + + obj.A = A; + obj.B = B; + obj.C = C; + obj.G = G; + + AA = 1/k^2*A + 1/2*B + 1/(2*k)*C; + BB = -2/k^2*A; + CC = 1/k^2*A + 1/2*B - 1/(2*k)*C; + % AA*v_next + BB*v + CC*v_prev == G(t_n) + + obj.AA = AA; + obj.BB = BB; + obj.CC = CC; + + v_prev = f1; + I = speye(m); + v = (1/k^2*A)\((1/k^2*A - 1/2*B)*f1 + (1/k*I - 1/2*C)*f2 + 1/2*G(0)); + + if ~issparse(A) || ~issparse(B) || ~issparse(C) + error('LU factorization with full pivoting only works for sparse matrices.') + end + + [L,U,p,q] = lu(AA,'vector'); + + obj.L = L; + obj.U = U; + obj.p = p; + obj.q = q; + + + obj.k = k; + obj.t = t0+k; + obj.n = 1; + obj.v = v; + obj.v_prev = v_prev; + end + + function [v,t] = getV(obj) + v = obj.v; + t = obj.t; + end + + function [vt,t] = getVt(obj) + % Calculate next time step to be able to do centered diff. + v_next = zeros(size(obj.v)); + b = obj.G(obj.t) - obj.BB*obj.v - obj.CC*obj.v_prev; + + y = obj.L\b(obj.p); + z = obj.U\y; + v_next(obj.q) = z; + + + vt = (v_next-obj.v_prev)/(2*obj.k); + t = obj.t; + end + + function obj = step(obj) + b = obj.G(obj.t) - obj.BB*obj.v - obj.CC*obj.v_prev; + obj.v_prev = obj.v; + + % % Backslash + % obj.v = obj.AA\b; + + % LU with column pivot + y = obj.L\b(obj.p); + z = obj.U\y; + obj.v(obj.q) = z; + + % Update time + obj.t = obj.t + obj.k; + obj.n = obj.n + 1; + end + end +end + + + + + +%%% Derivation +% syms A B C G +% syms n k +% syms f1 f2 + +% v = symfun(sym('v(n)'),n); + + +% d = A/k^2 * (v(n+1) - 2*v(n) +v(n-1)) + B/2*(v(n+1)+v(n-1)) + C/(2*k)*(v(n+1) - v(n-1)) == G +% ic1 = v(0) == f1 +% ic2 = A/k*(v(1)-f1) + k/2*(B*f1 + C*f2 - G) - f2 == 0 + +% c = collect(d, [v(n) v(n-1) v(n+1)]) % (-(2*A)/k^2)*v(n) + (B/2 + A/k^2 - C/(2*k))*v(n - 1) + (B/2 + A/k^2 + C/(2*k))*v(n + 1) == G +% syms AA BB CC +% % AA = B/2 + A/k^2 + C/(2*k) +% % BB = -(2*A)/k^2 +% % CC = B/2 + A/k^2 - C/(2*k) +% s = subs(c, [B/2 + A/k^2 + C/(2*k), -(2*A)/k^2, B/2 + A/k^2 - C/(2*k)], [AA, BB, CC]) + + +% ic2_a = collect(ic2, [v(1) f1 f2]) % (A/k)*v(1) + ((B*k)/2 - A/k)*f1 + ((C*k)/2 - 1)*f2 - (G*k)/2 == 0 +
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/CdiffTimeDep.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+time/CdiffTimeDep.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,51 @@ +classdef CdiffTimeDep < time.Timestepper + properties + D + E + S + k + t + v + v_prev + n + end + + + methods + % Solves u_tt = Du + E(t)u_t + S(t) + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. + % CdiffTimeDep(D, E, S, k, t0, n0, v, v_prev) + function obj = CdiffTimeDep(D, E, S, k, t0, n0, v, v_prev) + m = length(v); + default_arg('E', @(t)sparse(m,m)); + default_arg('S', @(t)sparse(m,1)); + + obj.D = D; + obj.E = E; + obj.S = S; + + obj.k = k; + obj.t = t0; + obj.n = n0; + obj.v = v; + obj.v_prev = v_prev; + end + + function [v,t] = getV(obj) + v = obj.v; + t = obj.t; + end + + function [vt,t] = getVt(obj) + vt = (obj.v-obj.v_prev)/obj.k; % Could be improved using u_tt = f(u)) + t = obj.t; + end + + function obj = step(obj) + [obj.v, obj.v_prev] = time.cdiff.cdiff(obj.v, obj.v_prev, obj.k, obj.D, obj.E(obj.t), obj.S(obj.t)); + obj.t = obj.t + obj.k; + obj.n = obj.n + 1; + end + end +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/Rungekutta4SecondOrder.m --- a/+time/Rungekutta4SecondOrder.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+time/Rungekutta4SecondOrder.m Tue Feb 07 16:09:02 2017 +0100 @@ -15,6 +15,9 @@ methods + % Solves u_tt = Du + Eu_t + S + % D, E, S can either all be constants or all be function handles, + % They can also be omitted by setting them equal to the empty matrix. function obj = Rungekutta4SecondOrder(D, E, S, k, t0, v0, v0t) obj.D = D; obj.E = E; @@ -22,21 +25,63 @@ obj.m = length(v0); obj.n = 0; - I = speye(obj.m); - O = sparse(obj.m,obj.m); - obj.M = [O, I; D, E*I]; % Multiply with I to allow 0 as input. + + if isa(D, 'function_handle') || isa(E, 'function_handle') || isa(S, 'function_handle') + default_arg('D', @(t)sparse(obj.m, obj.m)); + default_arg('E', @(t)sparse(obj.m, obj.m)); + default_arg('S', @(t)sparse(obj.m, 1) ); + + if ~isa(D, 'function_handle') + D = @(t)D; + end + if ~isa(E, 'function_handle') + E = @(t)E; + end + if ~isa(S, 'function_handle') + S = @(t)S; + end - if S == 0 - obj.C = zeros(2*obj.m,1); + I = speye(obj.m); + O = sparse(obj.m,obj.m); + + obj.M = @(t)[ + O, I; + D(t), E(t); + ]; + obj.C = @(t)[ + zeros(obj.m,1); + S(t); + ]; + + obj.k = k; + obj.t = t0; + obj.w = [v0; v0t]; + + obj.F = @(w,t)(obj.M(t)*w + obj.C(t)); else - obj.C = [zeros(obj.m,1), S]; - end + + default_arg('D', sparse(obj.m, obj.m)); + default_arg('E', sparse(obj.m, obj.m)); + default_arg('S', sparse(obj.m, 1) ); + + I = speye(obj.m); + O = sparse(obj.m,obj.m); - obj.k = k; - obj.t = t0; - obj.w = [v0; v0t]; + obj.M = [ + O, I; + D, E; + ]; + obj.C = [ + zeros(obj.m,1); + S; + ]; - obj.F = @(w,t)(obj.M*w + obj.C); + obj.k = k; + obj.t = t0; + obj.w = [v0; v0t]; + + obj.F = @(w,t)(obj.M*w + obj.C); + end end function [v,t] = getV(obj)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/Rungekutta4proper.m --- a/+time/Rungekutta4proper.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+time/Rungekutta4proper.m Tue Feb 07 16:09:02 2017 +0100 @@ -10,6 +10,7 @@ methods + % Timesteps v_t = F(v,t), using RK4 fromt t = t0 with timestep k and initial conditions v = v0 function obj = Rungekutta4proper(F, k, t0, v0) obj.F = F; obj.k = k;
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/SBPInTime.m --- a/+time/SBPInTime.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+time/SBPInTime.m Tue Feb 07 16:09:02 2017 +0100 @@ -104,6 +104,7 @@ end end + methods(Static) function N = smallestBlockSize(order,TYPE) default_arg('TYPE','gauss')
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +time/Timestepper.m --- a/+time/Timestepper.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+time/Timestepper.m Tue Feb 07 16:09:02 2017 +0100 @@ -60,6 +60,13 @@ s = util.replace_string(s,''); end + + function [v, t] = stepTo(obj, n, progress_bar) + default_arg('progress_bar',false); + + [v, t] = obj.stepN(n-obj.n, progress_bar); + end + function [v,t] = evolve(obj, tend, progress_bar) default_arg('progress_bar',false) if ~progress_bar
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +util/ReplaceableString.m --- a/+util/ReplaceableString.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+util/ReplaceableString.m Tue Feb 07 16:09:02 2017 +0100 @@ -34,10 +34,20 @@ obj.display(); end + function show(obj, varargin) + obj.param = varargin; + obj.display(); + end + + function remove(obj) + obj.update(''); + end + function display(obj) reverseStr = repmat(sprintf('\b'), 1, obj.n); - newStr = padStr(sprintf(obj.fmt, obj.param{:}),obj.n); - fprintf([reverseStr, newStr]); + cleareStr = repmat(sprintf(' '), 1, obj.n); + newStr = sprintf(obj.fmt, obj.param{:}); + fprintf([reverseStr, cleareStr, reverseStr, newStr]); obj.n = length(newStr); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 +util/calc_borrowing.m --- a/+util/calc_borrowing.m Tue Feb 07 15:47:51 2017 +0100 +++ b/+util/calc_borrowing.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,97 +1,241 @@ +function calc_borrowing(m, h) + default_arg('m',100); + default_arg('h',1); -m = 100; -h = 1; + operators = { + { + 'd4_lonely', getM4_lonely, { + {4, 'min_boundary_points'}, + {6, 'min_boundary_points'}, + {6, '2'}, + {6, '3'}, + {8, 'min_boundary_points'}, + {8, 'higher_boundary_order'}, + } + }, { + 'd4_variable', { + {2}, + {4}, + {6}, + } + } + % BORKEN BAD IDEA + } + + + for i = 1:operators + baseName = operators{i}{1}; + postFixes = operators{i}{2}; + for pf = postFixes + [a2, a3] = borrowFromD4(m, h, l{:}); + end + end -%% 4th order non-compatible -[H, HI, D1, D2, D3, D4, e_1, e_m, M, M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher4(m,h); -S1 = S_1*S_1' + S_m*S_m'; -S2 = S2_1*S2_1' + S2_m*S2_m'; -S3 = S3_1*S3_1' + S3_m*S3_m'; + + lonely = { + {4, 'min_boundary_points'}, + {6, 'min_boundary_points'}, + {6, '2'}, + {6, '3'}, + {8, 'min_boundary_points'}, + {8, 'higher_boundary_order'}, + }; -alpha_I = util.matrixborrow(M4, h^-1*S1 ); -alpha_II = util.matrixborrow(M4, h*S2 ); -alpha_III = util.matrixborrow(M4, h^3*S3); -fprintf('4th order non-compatible\n') -fprintf('alpha_I1: %.10f\n',alpha_I) -fprintf('alpha_II: %.10f\n',alpha_II) -fprintf('alpha_III: %.10f\n',alpha_III) -fprintf('\n') - + for i = 1:length(lonely) + l = lonely{i}; + [a2, a3] = d4_lonely(m, h, l{:}); + fprintf('d4_lonely %d %s\n', l{:}) + fprintf('\t alpha_II = %f\n', a2) + fprintf('\t alpha_III = %f\n', a3) + fprintf('\n') + end -%% 6th order non-compatible -[H, HI, D1, D2, D3, D4, e_1, e_m, M, M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher6(m,h); -S1 = S_1*S_1' + S_m*S_m'; -S2 = S2_1*S2_1' + S2_m*S2_m'; -S3 = S3_1*S3_1' + S3_m*S3_m'; + variable = { + {2}, + {4}, + {6}, + }; -alpha_II = util.matrixborrow(M4, h*S2 ); -alpha_III = util.matrixborrow(M4, h^3*S3); -fprintf('6th order non-compatible\n') -fprintf('alpha_II: %.10f\n',alpha_II) -fprintf('alpha_III: %.10f\n',alpha_III) -fprintf('\n') + for i = 1:length(variable) + l = variable{i}; + [a2, a3] = d4_variable(m, h, l{:}); + fprintf('d4_variable %d\n', l{:}) + fprintf('\t alpha_II = %f\n', a2) + fprintf('\t alpha_III = %f\n', a3) + fprintf('\n') + end -%% 2nd order compatible -[H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible2(m,h); -S1 = S_1*S_1' + S_m*S_m'; -S2 = S2_1*S2_1' + S2_m*S2_m'; -S3 = S3_1*S3_1' + S3_m*S3_m'; + %% 4th order non-compatible + [H, HI, D1, D2, D3, D4, e_1, e_m, M, M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher4(m,h); + S1 = S_1*S_1' + S_m*S_m'; + S2 = S2_1*S2_1' + S2_m*S2_m'; + S3 = S3_1*S3_1' + S3_m*S3_m'; -alpha_II = util.matrixborrow(M4, h*S2 ); -alpha_III = util.matrixborrow(M4, h^3*S3); -fprintf('2nd order compatible\n') -fprintf('alpha_II: %.10f\n',alpha_II) -fprintf('alpha_III: %.10f\n',alpha_III) -fprintf('\n') + alpha_I = util.matrixborrow(M4, h^-1*S1 ); + alpha_II = util.matrixborrow(M4, h*S2 ); + alpha_III = util.matrixborrow(M4, h^3*S3); + fprintf('4th order non-compatible\n') + fprintf('alpha_I1: %.10f\n',alpha_I) + fprintf('alpha_II: %.10f\n',alpha_II) + fprintf('alpha_III: %.10f\n',alpha_III) + fprintf('\n') + + + %% 6th order non-compatible + [H, HI, D1, D2, D3, D4, e_1, e_m, M, M4,Q, Q3, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher6(m,h); + S1 = S_1*S_1' + S_m*S_m'; + S2 = S2_1*S2_1' + S2_m*S2_m'; + S3 = S3_1*S3_1' + S3_m*S3_m'; + + alpha_II = util.matrixborrow(M4, h*S2 ); + alpha_III = util.matrixborrow(M4, h^3*S3); + fprintf('6th order non-compatible\n') + fprintf('alpha_II: %.10f\n',alpha_II) + fprintf('alpha_III: %.10f\n',alpha_III) + fprintf('\n') -%% 4th order compatible -[H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible4(m,h); -S1 = S_1*S_1' + S_m*S_m'; -S2 = S2_1*S2_1' + S2_m*S2_m'; -S3 = S3_1*S3_1' + S3_m*S3_m'; + %% 2nd order compatible + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible2(m,h); + S1 = S_1*S_1' + S_m*S_m'; + S2 = S2_1*S2_1' + S2_m*S2_m'; + S3 = S3_1*S3_1' + S3_m*S3_m'; -alpha_II = util.matrixborrow(M4, h*S2 ); -alpha_III = util.matrixborrow(M4, h^3*S3); -fprintf('4th order compatible\n') -fprintf('alpha_II: %.10f\n',alpha_II) -fprintf('alpha_III: %.10f\n',alpha_III) -fprintf('\n') + alpha_II = util.matrixborrow(M4, h*S2 ); + alpha_III = util.matrixborrow(M4, h^3*S3); + fprintf('2nd order compatible\n') + fprintf('alpha_II: %.10f\n',alpha_II) + fprintf('alpha_III: %.10f\n',alpha_III) + fprintf('\n') + + + %% 4th order compatible + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible4(m,h); + S1 = S_1*S_1' + S_m*S_m'; + S2 = S2_1*S2_1' + S2_m*S2_m'; + S3 = S3_1*S3_1' + S3_m*S3_m'; -%% 6th order compatible -[H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible6(m,h); -S1 = S_1*S_1' + S_m*S_m'; -S2 = S2_1*S2_1' + S2_m*S2_m'; -S3 = S3_1*S3_1' + S3_m*S3_m'; + alpha_II = util.matrixborrow(M4, h*S2 ); + alpha_III = util.matrixborrow(M4, h^3*S3); + fprintf('4th order compatible\n') + fprintf('alpha_II: %.10f\n',alpha_II) + fprintf('alpha_III: %.10f\n',alpha_III) + fprintf('\n') -alpha_II = util.matrixborrow(M4, h*S2 ); -alpha_III = util.matrixborrow(M4, h^3*S3); -fprintf('6th order compatible\n') -fprintf('alpha_II: %.10f\n',alpha_II) -fprintf('alpha_III: %.10f\n',alpha_III) -fprintf('\n') + %% 6th order compatible + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher_compatible6(m,h); + S1 = S_1*S_1' + S_m*S_m'; + S2 = S2_1*S2_1' + S2_m*S2_m'; + S3 = S3_1*S3_1' + S3_m*S3_m'; + + alpha_II = util.matrixborrow(M4, h*S2 ); + alpha_III = util.matrixborrow(M4, h^3*S3); + fprintf('6th order compatible\n') + fprintf('alpha_II: %.10f\n',alpha_II) + fprintf('alpha_III: %.10f\n',alpha_III) + fprintf('\n') -% Ordinary + % Ordinary + + for order = [2 4 6 8 10] + op = sbp.Ordinary(m,h, order); + + S_1 = op.boundary.S_1; + S_m = op.boundary.S_m; + + M = op.norms.M; -for order = [2 4 6 8 10] - op = sbp.Ordinary(m,h, order); + S1 = S_1*S_1' + S_m*S_m'; + alpha = util.matrixborrow(M, h*S1); + fprintf('%dth order Ordinary\n', order) + fprintf('alpha: %.10f\n', alpha) + fprintf('\n') + end - S_1 = op.boundary.S_1; - S_m = op.boundary.S_m; + + + +end - M = op.norms.M; +function [alpha_II, alpha_III] = d4_lonely(m, h, order, modifier) + default_arg('modifier', []) + func = sprintf('sbp.implementations.d4_lonely_%d', order); + if ~isempty(modifier) + func = sprintf('%s_%s', func, modifier); + end + funcCall = sprintf('%s(%s,%s)', func, toString(m), toString(h)); + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = eval(funcCall); + + d2d2 = d2_l*d2_l' + d2_r*d2_r'; + alpha_II = util.matrixborrow(M4, h*d2d2); + + d3d3 = d3_l*d3_l' + d3_r*d3_r'; + alpha_III = util.matrixborrow(M4, h^3*d3d3); +end - S1 = S_1*S_1' + S_m*S_m'; - alpha = util.matrixborrow(M, h*S1); - fprintf('%dth order Ordinary\n', order) - fprintf('alpha: %.10f\n', alpha) - fprintf('\n') +function [alpha_II, alpha_III] = d4_variable(m, h, order) + default_arg('modifier', []) + func = sprintf('sbp.implementations.d4_variable_%d', order); + + funcCall = sprintf('%s(%s,%s)', func, toString(m), toString(h)); + [H, HI, D1, D2, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = eval(funcCall); + + d2d2 = d2_l*d2_l' + d2_r*d2_r'; + alpha_II = util.matrixborrow(M4, h*d2d2); + + d3d3 = d3_l*d3_l' + d3_r*d3_r'; + alpha_III = util.matrixborrow(M4, h^3*d3d3); +end + +function [d2_l, d2_r, d3_l, d3_r, M4] = getM4_lonely(m, h, order, modifier) + fStr = getFunctionCallStr('d4_lonely', {order, modifier}, {m ,h}); + [H, HI, D4, e_l, e_r, M4, d2_l, d2_r, d3_l, d3_r, d1_l, d1_r] = eval(funcCall); end +% Calculates the borrowing constants for a D4 operator. +% getM4 is a function handle on the form +% [d2_l, d2_r, d3_l, d3_r, M4] = getM4(m,h) +function [a2, a3] = borrowFromD4(m, h, getM4) + [d2_l, d2_r, d3_l, d3_r, M4] = getM4(m, h); + + d2d2 = d2_l*d2_l' + d2_r*d2_r'; + a2 = util.matrixborrow(M4, h*d2d2); + + d3d3 = d3_l*d3_l' + d3_r*d3_r'; + a3 = util.matrixborrow(M4, h^3*d3d3); +end + + +function funcCallStr = getFunctionCallStr(baseName, postFix, parameters) + default_arg('postFix', []) + default_arg('parameters', []) + + funcCallStr = sprintf('sbp.implementations.%s', baseName); + + for i = 1:length(postFix) + if ischar(postFix{i}) + funcCallStr = [funcCallStr '_' postFix{i}]; + else + funcCallStr = [funcCallStr '_' toString(postFix{i})]; + end + end + + if isempty(parameters) + return + end + + funcCallStr = [funcCallStr '(' toString(parameters{1})]; + + for i = 2:length(parameters) + funcCallStr = [funcCallStr ', ' toString(parameters{i})]; + end + + funcCallStr = [funcCallStr ')'; +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 Dictionary.m --- a/Dictionary.m Tue Feb 07 15:47:51 2017 +0100 +++ b/Dictionary.m Tue Feb 07 16:09:02 2017 +0100 @@ -98,14 +98,11 @@ % Should probably use mat2str with some kind of normalization to make all variables valied fieldname % and make it possible to recover the value function fName = getFieldname(obj, val) - if isnumeric(val) - valStr = num2str(val); - elseif ischar(val) - valStr = val; - else - error('Dont know what to do with val!'); + if ~ischar(val) + val = toString(val); end - fName = sprintf('f%s',valStr); + + fName = matlab.lang.makeValidName(val); end end end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 centerColorbar.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/centerColorbar.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,6 @@ +function centerColorbar(ah) + old = ah.CLim; + + l = max(abs(old)); + ah.CLim = [-l l]; +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 copyWithDefault.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/copyWithDefault.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,31 @@ +% Copy the struct src to dest with default values from default +% dest = copyWithDefault(src, default) +function dest = copyWithDefault(src, default) + % src does not have a value => use default + if isempty(src) + dest = default; + return + end + + % src has a value and is not a struct => use src + % src has a value and default is not a struct => use src + if ~isstruct(src) || ~isstruct(default) + dest = src; + return + end + + + % src has a value and is a struct => add all default fields + dest = src; + + fn = fieldnames(default); + for i = 1:length(fn) + if isfield(src, fn{i}) + srcField = src.(fn{i}); + else + srcField = []; + end + + dest.(fn{i}) = copyWithDefault(srcField, default.(fn{i})); + end +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 default_struct.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/default_struct.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,10 @@ +function default_struct(s, val) + if evalin('caller',sprintf('~exist(''%s'',''var'')',s)) + given = []; + else + given = evalin('caller', s); + end + + final = copyWithDefault(given, val); + assignin('caller', s, final); +end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 findZeros.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/findZeros.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,30 @@ +% findZeros looks for solutions to the equation f(x)==0 within +% the limits lim with a granularity of h. +% Returns a sorted list of unique solutions. +function z = findZeros(f, lim, h) + n = ceil((lim(2)-lim(1))/h); + z0 = linspace(lim(1), lim(2), n); + + z = zeros(1,n); + + for i = 1:n + zt(i) = fzero(f, z0(i)); + end + + zt = sort(zt); + + z = []; + for i = 1:n + if zt(i) < lim(1) || zt(i) > lim(2) + continue + end + + if ~isempty(z) && abs(z(end) - zt(i)) < 1e-6 + continue + end + + z = [z zt(i)]; + end + + % z = unique(z); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 minors.m --- a/minors.m Tue Feb 07 15:47:51 2017 +0100 +++ b/minors.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,4 +1,5 @@ -function [minor, sub] = minors(A) +function [minor, sub] = minors(A, verbose) + default_arg('verbose', true); [n, m] = size(A); if n ~= m @@ -18,8 +19,10 @@ end for i = 1:length(sub) - fprintf('%d:\n', ks{i}); - disp(sub{i}) + if verbose + fprintf('%d:\n', ks{i}); + disp(sub{i}) + end minor(i) = det(sub{i}); end
diff -r a2cb0d4f4a02 -r e56dbd9e4196 saveFigureSize.m --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/saveFigureSize.m Tue Feb 07 16:09:02 2017 +0100 @@ -0,0 +1,6 @@ +function saveFigurePosition() + defaultPosition = get(0,'defaultfigureposition'); + f = gcf; + defaultPosition(3:4) = f.Position(3:4); + set(0,'defaultfigureposition',defaultPosition); +end \ No newline at end of file
diff -r a2cb0d4f4a02 -r e56dbd9e4196 semiDefIneq.m --- a/semiDefIneq.m Tue Feb 07 15:47:51 2017 +0100 +++ b/semiDefIneq.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,6 @@ -function ineq = semiDefIneq(A) - [m, sub] = minors(A); +function ineq = semiDefIneq(A, verbose) + default_arg('verbose', true); + [m, sub] = minors(A, verbose); ineqsys = true; for i = 1:length(m)
diff -r a2cb0d4f4a02 -r e56dbd9e4196 timeTask.m --- a/timeTask.m Tue Feb 07 15:47:51 2017 +0100 +++ b/timeTask.m Tue Feb 07 16:09:02 2017 +0100 @@ -1,5 +1,5 @@ -function done = timeTask(taskName) - fprintf('%s', taskName); +function done = timeTask(fmt, varargin) + fprintf(fmt, varargin{:}); tStart = tic; function done_fun()