Mercurial > repos > public > sbplib
changeset 755:14f0058356f2 feature/d1_staggered
Merge with feature/grids
| author | Martin Almquist <malmquist@stanford.edu> |
|---|---|
| date | Fri, 15 Jun 2018 18:10:26 -0700 |
| parents | 5264ce57b573 (diff) 87436a107d8a (current diff) |
| children | f891758ad7a4 |
| files | |
| diffstat | 18 files changed, 1972 insertions(+), 0 deletions(-) [+] |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/Staggered1d.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,99 @@ +classdef Staggered1d < grid.Structured + properties + grids % Cell array of grids + Ngrids % Number of grids + h % Interior grid spacing + d % Number of dimensions + end + + methods + + % Accepts multiple grids and combines them into a staggered grid + function obj = Staggered1d(varargin) + + obj.d = 1; + + obj.Ngrids = length(varargin); + obj.grids = cell(obj.Ngrids, 1); + for i = 1:obj.Ngrids + obj.grids{i} = varargin{i}; + end + + obj.h = []; + end + + % N returns the total number of points + function o = N(obj) + o = 0; + for i=1:obj.Ngrids + o = o+obj.grids{i}.size(); + end + end + + % D returns the spatial dimension of the grid + function o = D(obj) + o = obj.d; + end + + % size returns the number of points on the primal grid + function m = size(obj) + m = obj.grids{1}.size(); + end + + % points returns an n x 1 vector containing the coordinates for all points. + function X = points(obj) + X = []; + for i = 1:obj.Ngrids + X = [X; obj.grids{i}.points()]; + end + end + + % matrices returns a cell array with coordinates in matrix form. + % For 2d case these will have to be transposed to work with plotting routines. + function X = matrices(obj) + + % There is no 1d matrix data type in matlab, handle special case + X{1} = reshape(obj.points(), [obj.m 1]); + + end + + function h = scaling(obj) + if isempty(obj.h) + error('grid:Staggered1d:NoScalingSet', 'No scaling set') + end + + h = obj.h; + end + + % Restricts the grid function gf on obj to the subgrid g. + % Only works for even multiples + function gf = restrictFunc(obj, gf, g) + error('grid:Staggered1d:NotImplemented','This method does not exist yet') + end + + % Projects the grid function gf on obj to the grid g. + function gf = projectFunc(obj, gf, g) + error('grid:Staggered1d:NotImplemented','This method does not exist yet') + end + + % Return the names of all boundaries in this grid. + function bs = getBoundaryNames(obj) + switch obj.D() + case 1 + bs = {'l', 'r'}; + case 2 + bs = {'w', 'e', 's', 'n'}; + case 3 + bs = {'w', 'e', 's', 'n', 'd', 'u'}; + otherwise + error('not implemented'); + end + end + + % Return coordinates for the given boundary + function X = getBoundary(obj, name) + % Use boundaries of first grid + X = obj.grids{1}.getBoundary(name); + end + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/primalDual1D.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,30 @@ +% Creates a 1D staggered grid of dimension length(m). +% over the interval xlims +% Primal grid: equidistant with m points. +% Dual grid: m + 1 points, h/2 spacing first point. +% Examples +% g = grid.primal_dual_1D(m, xlim) +% g = grid.primal_dual_1D(11, {0,1}) +function [g_primal, g_dual] = primalDual1D(m, xlims) + + if ~iscell(xlims) || numel(xlims) ~= 2 + error('grid:primalDual1D:InvalidLimits','The limits should be cell arrays with 2 elements.'); + end + + if xlims{1} > xlims{2} + error('grid:primalDual1D:InvalidLimits','The elements of the limit must be increasing.'); + end + + xl = xlims{1}; + xr = xlims{2}; + h = (xr-xl)/(m-1); + + % Primal grid + g_primal = grid.equidistant(m, xlims); + g_primal.h = h; + + % Dual grid + x = [xl; linspace(xl+h/2, xr-h/2, m-1)'; xr]; + g_dual = grid.Cartesian(x); + g_dual.h = h; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+grid/primalDual1DTest.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,32 @@ +function tests = primalDual1DTest() + tests = functiontests(localfunctions); +end + + +function testErrorInvalidLimits(testCase) + in = { + {10,{1}}, + {10,[0,1]}, + {10,{1,0}}, + }; + + for i = 1:length(in) + testCase.verifyError(@()grid.primalDual1D(in{i}{:}),'grid:primalDual1D:InvalidLimits',sprintf('in(%d) = %s',i,toString(in{i}))); + end +end + +function testCompiles(testCase) + in = { + {5, {0,1}}, + }; + + out = { + {[0; 0.25; 0.5; 0.75; 1], [0; 0.125; 0.375; 0.625; 0.875; 1]}, + }; + + for i = 1:length(in) + [gp, gd] = grid.primalDual1D(in{i}{:}); + testCase.verifyEqual(gp.points(),out{i}{1}); + testCase.verifyEqual(gd.points(),out{i}{2}); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_2.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,71 @@ +function [xp, xm, Pp, Pm, Qp, Qm] = d1_staggered_2(n, L) +% [xp, xm, Pp, Pm, Qp, Qm] = sbp_staggered_2(n, L) +% Input arguments: +% n : number of grid points: n on xp grid and n+1 on xm grid +% L : interval length + +h = L/(n-1); + +n = n - 1; +assert(n >= 3, 'Not enough grid points'); + +qm20 = -1/4; +qm01 = 1/2; +qp11 = -1/4; +qm11 = 1/4; +qp02 = 1/4; +pp0 = 1/2; +pm1 = 1/4; +qm00 = -1/2; +qp10 = -1/2; +qp12 = 3/4; +qp00 = -1/2; +qm21 = -3/4; +pp1 = 1; +qp01 = 1/4; +pm0 = 1/2; +pm2 = 5/4; +qm10 = -1/4; + +% Q+ and Q-, top-left corner +QpL = [... +qp00, qp01, qp02; + qp10, qp11, qp12 +]; +QmL = [... +qm00, qm01; + qm10, qm11; + qm20, qm21 +]; + +% Q+ and Q- +b = 2; +w = b; +s = [-1, 1]'; +Qp = spdiags(repmat(-s(end:-1:1)',[n+2 1]), -(w/2-1):w/2, n+2, n+2); +Qm = spdiags(repmat(s(:)',[n+2 1]), -(w/2-1)-1:w/2-1, n+2, n+2); +Qp(end,:) = []; +Qm(:,end) = []; + +% Add SBP boundary closures +Qp(1:b,1:b+1) = QpL; +Qp(end-b+1:end,end-b:end) = -fliplr(flipud(QpL)); +Qm(1:b+1,1:b) = QmL; +Qm(end-b:end,end-b+1:end) = -fliplr(flipud(QmL)); + +% P+ and P- +Pp = ones(n+1,1); +Pm = ones(n+2,1); + +Pp(1:b) = [pp0, pp1]; +Pp(end-b+1:end) = Pp(b:-1:1); +Pm(1:b+1) = [pm0, pm1, pm2]; +Pm(end-b:end) = Pm(b+1:-1:1); +Pp = spdiags(Pp,0,n+1,n+1); +Pm = spdiags(Pm,0,n+2,n+2); + +Pp = h*Pp; +Pm = h*Pm; + +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_4.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,58 @@ +function [xp, xm, Pp, Pm, Qp, Qm] = d1_staggered_4(n, L) +% [xp, xm, Pp, Pm, Qp, Qm] = sbp_staggered_4(n, L) +% Input arguments: +% n : number of grid points: n on xp grid and n+1 on xm grid +% L : interval length + +h = L/(n-1); + +n = n - 1; +assert(n >= 7, 'Not enough grid points'); + +% Q+ and Q-, top-left corner +QpL = [... +-323/378, 2783/3072, -1085/27648, -17/9216, -667/64512; + -5/21, -847/1024, 1085/1024, -37/1024, 899/21504; + 5/42, -121/1024, -1085/1024, 3575/3072, -753/7168; + -5/189, 121/3072, 1085/27648, -10759/9216, 74635/64512 +]; +QmL = [... +-55/378, 5/21, -5/42, 5/189; + -2783/3072, 847/1024, 121/1024, -121/3072; + 1085/27648, -1085/1024, 1085/1024, -1085/27648; + 17/9216, 37/1024, -3575/3072, 10759/9216; + 667/64512, -899/21504, 753/7168, -74635/64512 +]; + +% Q+ and Q- +w = 4; +s = [ 1/24, -9/8, 9/8, -1/24]; +Qp = spdiags(repmat(-s(end:-1:1),[n+2 1]), -(w/2-1):w/2, n+2, n+2); +Qm = spdiags(repmat(s(:)',[n+2 1]), -(w/2-1)-1:w/2-1, n+2, n+2); +Qp(end,:) = []; +Qm(:,end) = []; + +% Add SBP boundary closures +bp = 4; +bm = 5; +Qp(1:bp,1:bm) = double(QpL); +Qp(end-bp+1:end,end-bm+1:end) = -fliplr(flipud(QpL)); +Qm(1:bm,1:bp) = double(QmL); +Qm(end-bm+1:end,end-bp+1:end) = -fliplr(flipud(QmL)); + +% P+ and P- +Pp = ones(n+1,1); +Pm = ones(n+2,1); + +Pp(1:bp) = [407/1152, 473/384, 343/384, 1177/1152]; +Pp(end-bp+1:end) = Pp(bp:-1:1); +Pm(1:bm) = [5/63, 121/128, 1085/1152, 401/384, 2659/2688]; +Pm(end-bm+1:end) = Pm(bm:-1:1); +Pp = spdiags(Pp,0,n+1,n+1); +Pm = spdiags(Pm,0,n+2,n+2); + +Pp = h*Pp; +Pm = h*Pm; + +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_6.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,103 @@ +function [xp, xm, Pp, Pm, Qp, Qm] = d1_staggered_6(n, L) +% [xp, xm, Pp, Pm, Qp, Qm] = sbp_staggered_6(n, L) +% Input arguments: +% n : number of grid points: n on xp grid and n+1 on xm grid +% L : interval length + +h = L/(n-1); + +n = n - 1; +assert(n >= 9, 'Not enough grid points'); + +% Coefficients determined such that the SBP property is satisfied +qp44 = -144662434693018337/131093333881896960; +pp3 = 793/720; +qp22 = -729260819980499/780317463582720; +pm5 = 1262401/1244160; +pp4 = 157/160; +qp03 = -1056618403248889/17835827739033600; +qp32 = 17518644727427/2340952390748160; +pm4 = 224201/241920; +pm2 = 90101/103680; +qp40 = -572605772036807/27311111225395200; +qp30 = 3572646778087103/61450000257139200; +pm3 = 26369/23040; +qp41 = 935926732952051/29964190601576448; +qp02 = -67953322736279/2340952390748160; +pp0 = 95/288; +qp01 = 143647626126052207/149820953007882240; +pp1 = 317/240; +qp12 = 444281115722813/468190478149632; +qp25 = 312883915256963/24970158834647040; +qp11 = -36400588465707331/37455238251970560; +qp20 = -137012145866261/13655555612697600; +qp10 = -1008199398207353/6827777806348800; +qp15 = -1088192686015471/37455238251970560; +qp04 = -239061311631881/131093333881896960; +qp35 = -2480203967217637/112365714755911680; +qp21 = 1658590350384343/24970158834647040; +qp05 = 4843362440877649/449462859023646720; +qp24 = -372310079307409/4369777796063232; +qp33 = -4809914536549211/4458956934758400; +qp23 = 2826535075774271/2972637956505600; +qp42 = 4202889834071/585238097687040; +qp14 = 492703999899311/32773333470474240; +pm1 = 133801/138240; +qp34 = 36544940032227659/32773333470474240; +qp00 = -216175739231516543/245800001028556800; +qp43 = 128340370711031/17835827739033600; +qp13 = 823082791653949/4458956934758400; +pm0 = 5251/68040; +pp2 = 23/30; +qp45 = 170687698457070187/149820953007882240; +qp31 = -3169112007572299/37455238251970560; + +% Q+ and Q-, top-left corner +QpL = [... +-216175739231516543/245800001028556800, 143647626126052207/149820953007882240, -67953322736279/2340952390748160, -1056618403248889/17835827739033600, -239061311631881/131093333881896960, 4843362440877649/449462859023646720; + -1008199398207353/6827777806348800, -36400588465707331/37455238251970560, 444281115722813/468190478149632, 823082791653949/4458956934758400, 492703999899311/32773333470474240, -1088192686015471/37455238251970560; + -137012145866261/13655555612697600, 1658590350384343/24970158834647040, -729260819980499/780317463582720, 2826535075774271/2972637956505600, -372310079307409/4369777796063232, 312883915256963/24970158834647040; + 3572646778087103/61450000257139200, -3169112007572299/37455238251970560, 17518644727427/2340952390748160, -4809914536549211/4458956934758400, 36544940032227659/32773333470474240, -2480203967217637/112365714755911680; + -572605772036807/27311111225395200, 935926732952051/29964190601576448, 4202889834071/585238097687040, 128340370711031/17835827739033600, -144662434693018337/131093333881896960, 170687698457070187/149820953007882240 +]; +QmL = [... +-29624261797040257/245800001028556800, 1008199398207353/6827777806348800, 137012145866261/13655555612697600, -3572646778087103/61450000257139200, 572605772036807/27311111225395200; + -143647626126052207/149820953007882240, 36400588465707331/37455238251970560, -1658590350384343/24970158834647040, 3169112007572299/37455238251970560, -935926732952051/29964190601576448; + 67953322736279/2340952390748160, -444281115722813/468190478149632, 729260819980499/780317463582720, -17518644727427/2340952390748160, -4202889834071/585238097687040; + 1056618403248889/17835827739033600, -823082791653949/4458956934758400, -2826535075774271/2972637956505600, 4809914536549211/4458956934758400, -128340370711031/17835827739033600; + 239061311631881/131093333881896960, -492703999899311/32773333470474240, 372310079307409/4369777796063232, -36544940032227659/32773333470474240, 144662434693018337/131093333881896960; + -4843362440877649/449462859023646720, 1088192686015471/37455238251970560, -312883915256963/24970158834647040, 2480203967217637/112365714755911680, -170687698457070187/149820953007882240 +]; + +% Q+ and Q- +w = 6; +s = [ -3/640, 25/384, -75/64, 75/64, -25/384, 3/640]; +Qp = spdiags(repmat(-s(end:-1:1),[n+2 1]), -(w/2-1):w/2, n+2, n+2); +Qm = spdiags(repmat(s(:)',[n+2 1]), -(w/2-1)-1:w/2-1, n+2, n+2); +Qp(end,:) = []; +Qm(:,end) = []; + +% Add SBP boundary closures +bp = 5; +bm = 6; +Qp(1:bp,1:bm) = double(QpL); +Qp(end-bp+1:end,end-bm+1:end) = -fliplr(flipud(QpL)); +Qm(1:bm,1:bp) = double(QmL); +Qm(end-bm+1:end,end-bp+1:end) = -fliplr(flipud(QmL)); + +% P+ and P- +Pp = ones(n+1,1); +Pm = ones(n+2,1); + +Pp(1:bp) = [95/288, 317/240, 23/30, 793/720, 157/160]; +Pp(end-bp+1:end) = Pp(bp:-1:1); +Pm(1:bm) = [5251/68040, 133801/138240, 90101/103680, 26369/23040, 224201/241920, 1262401/1244160]; +Pm(end-bm+1:end) = Pm(bm:-1:1); +Pp = spdiags(Pp,0,n+1,n+1); +Pm = spdiags(Pm,0,n+2,n+2); + +Pp = h*Pp; +Pm = h*Pm; + +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_upwind_2.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,73 @@ +function [xp,xm,Hp,Hm,HIp,HIm,Dp,Dm,Dpp,Dpm,Dmp,Dmm] = d1_staggered_upwind_2(n,L) + +assert(n-1 >= 6,'Not enough grid points'); + +np=n; +nm=n+1; +h = L/(n-1); + + +%H- norm +Hm_U=[0.88829e5 / 0.599299e6 0 0; 0 0.8056187e7 / 0.9588784e7 0; 0 0 0.9700117e7 / 0.9588784e7;]; +%H+ norm +Hp_U=[0.7489557e7 / 0.19177568e8 0 0; 0 0.1386089e7 / 0.1198598e7 0; 0 0 0.18276939e8 / 0.19177568e8;]; +% Upper part of Qpp. Notice that the B matrix is not included here. +% Qpp+Qpp^T=S/2,Qpp+Qpm^T=0 +Qpp_U=[-0.1e1 / 0.32e2 0.12886609e8 / 0.19177568e8 -0.1349263e7 / 0.9588784e7; -0.10489413e8 / 0.19177568e8 -0.3e1 / 0.16e2 0.16482403e8 / 0.19177568e8; 0.187491e6 / 0.2397196e7 -0.9290815e7 / 0.19177568e8 -0.11e2 / 0.32e2;]; +% Upper part of Qmp. Notice that the B matrix is not included here. +% Qmp+Qmp^T=S/2,Qmp+Qmm^T=0 +Qmp_U=[-0.2e1 / 0.21e2 0.12495263e8 / 0.16780372e8 -0.7520839e7 / 0.50341116e8; -0.7700871e7 / 0.16780372e8 -0.31e2 / 0.112e3 0.57771939e8 / 0.67121488e8; 0.2726447e7 / 0.50341116e8 -0.31402783e8 / 0.67121488e8 -0.113e3 / 0.336e3;]; +% The staggered + operator, upper part. Notice that the B matrix is not included here.Qp+Qm^T=0 +Qp_U=[-0.801195e6 / 0.2397196e7 0.16507959e8 / 0.19177568e8 -0.509615e6 / 0.19177568e8; -0.219745e6 / 0.1198598e7 -0.2112943e7 / 0.2397196e7 0.2552433e7 / 0.2397196e7; 0.42087e5 / 0.2397196e7 0.395585e6 / 0.19177568e8 -0.19909849e8 / 0.19177568e8;]; +Hp=spdiags(ones(np,1),0,np,np); +Hp(1:3,1:3)=Hp_U; +Hp(np-2:np,np-2:np)=fliplr(flipud(Hp_U)); +Hp=Hp*h; +HIp=inv(Hp); + +Hm=spdiags(ones(nm,1),0,nm,nm); +Hm(1:3,1:3)=Hm_U; +Hm(nm-2:nm,nm-2:nm)=fliplr(flipud(Hm_U)); +Hm=Hm*h; +HIm=inv(Hm); + +Qpp=spdiags(repmat([-0.3e1 / 0.8e1 -0.3e1 / 0.8e1 0.7e1 / 0.8e1 -0.1e1 / 0.8e1;],[np,1]),-1:2,np,np); +Qpp(1:3,1:3)=Qpp_U; +Qpp(np-2:np,np-2:np)=flipud( fliplr(Qpp_U(1:3,1:3) ) )'; + +Qpm=-Qpp'; + +Qmp=spdiags(repmat([-0.3e1 / 0.8e1 -0.3e1 / 0.8e1 0.7e1 / 0.8e1 -0.1e1 / 0.8e1;],[nm,1]),-1:2,nm,nm); +Qmp(1:3,1:3)=Qmp_U; +Qmp(nm-2:nm,nm-2:nm)=flipud( fliplr(Qmp_U(1:3,1:3) ) )'; + +Qmm=-Qmp'; + + +Bpp=spalloc(np,np,2);Bpp(1,1)=-1;Bpp(np,np)=1; +Bmp=spalloc(nm,nm,2);Bmp(1,1)=-1;Bmp(nm,nm)=1; + +Dpp=HIp*(Qpp+1/2*Bpp) ; +Dpm=HIp*(Qpm+1/2*Bpp) ; + + +Dmp=HIm*(Qmp+1/2*Bmp) ; +Dmm=HIm*(Qmm+1/2*Bmp) ; + + +%%% Start with the staggered +Qp=spdiags(repmat([-1 1],[np,1]),0:1,np,nm); +Qp(1:3,1:3)=Qp_U; +Qp(np-2:np,nm-2:nm)=flipud( fliplr(-Qp_U(1:3,1:3) ) ); +Qm=-Qp'; + +Bp=spalloc(np,nm,2);Bp(1,1)=-1;Bp(np,nm)=1; +Bm=Bp'; + +Dp=HIp*(Qp+1/2*Bp) ; + +Dm=HIm*(Qm+1/2*Bm) ; + +% grids +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_upwind_4.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,76 @@ +function [xp,xm,Hp,Hm,HIp,HIm,Dp,Dm,Dpp,Dpm,Dmp,Dmm] = d1_staggered_upwind_4(n,L) + +assert(n-1 >= 8,'Not enough grid points'); + +np=n; +nm=n+1; +h = L/(n-1); + + +%H- norm +Hm_U=[0.91e2 / 0.720e3 0 0 0; 0 0.325e3 / 0.384e3 0 0; 0 0 0.595e3 / 0.576e3 0; 0 0 0 0.1909e4 / 0.1920e4;]; +%H+ norm +Hp_U=[0.805e3 / 0.2304e4 0 0 0; 0 0.955e3 / 0.768e3 0 0; 0 0 0.677e3 / 0.768e3 0; 0 0 0 0.2363e4 / 0.2304e4;]; +% Upper part of Qpp. Notice that the B matrix is not included here. +% Qpp+Qpp^T=S/2,Qpp+Qpm^T=0 +Qpp_U=[-0.11e2 / 0.1536e4 0.1493e4 / 0.2304e4 -0.571e3 / 0.4608e4 -0.13e2 / 0.768e3; -0.697e3 / 0.1152e4 -0.121e3 / 0.1536e4 0.97e2 / 0.128e3 -0.473e3 / 0.4608e4; 0.373e3 / 0.4608e4 -0.139e3 / 0.256e3 -0.319e3 / 0.1536e4 0.1999e4 / 0.2304e4; 0.1e1 / 0.32e2 -0.121e3 / 0.4608e4 -0.277e3 / 0.576e3 -0.143e3 / 0.512e3;]; + +% Upper part of Qmp. Notice that the B matrix is not included here. +% Qmp+Qmp^T=S/2,Qmp+Qmm^T=0 +Qmp_U=[-0.209e3 / 0.5970e4 0.13439e5 / 0.17910e5 -0.13831e5 / 0.47760e5 0.10637e5 / 0.143280e6; -0.44351e5 / 0.71640e5 -0.20999e5 / 0.152832e6 0.230347e6 / 0.229248e6 -0.70547e5 / 0.254720e6; 0.3217e4 / 0.15920e5 -0.86513e5 / 0.114624e6 -0.15125e5 / 0.76416e5 0.1087241e7 / 0.1146240e7; -0.1375e4 / 0.28656e5 0.36117e5 / 0.254720e6 -0.655601e6 / 0.1146240e7 -0.211717e6 / 0.764160e6;]; + +% The staggered + operator, upper part. Notice that the B matrix is not included here.Qp+Qm^T=0 +Qp_U=[-0.338527e6 / 0.1004160e7 0.4197343e7 / 0.4819968e7 0.1423e4 / 0.803328e6 -0.854837e6 / 0.24099840e8; -0.520117e6 / 0.3012480e7 -0.492581e6 / 0.535552e6 0.2476673e7 / 0.2409984e7 0.520117e6 / 0.8033280e7; -0.50999e5 / 0.3012480e7 0.117943e6 / 0.1606656e7 -0.2476673e7 / 0.2409984e7 0.2712193e7 / 0.2677760e7; 0.26819e5 / 0.1004160e7 -0.117943e6 / 0.4819968e7 -0.1423e4 / 0.803328e6 -0.26119411e8 / 0.24099840e8;]; + +Hp=spdiags(ones(np,1),0,np,np); +Hp(1:4,1:4)=Hp_U; +Hp(np-3:np,np-3:np)=fliplr(flipud(Hp_U)); +Hp=Hp*h; +HIp=inv(Hp); + +Hm=spdiags(ones(nm,1),0,nm,nm); +Hm(1:4,1:4)=Hm_U; +Hm(nm-3:nm,nm-3:nm)=fliplr(flipud(Hm_U)); +Hm=Hm*h; +HIm=inv(Hm); + +Qpp=spdiags(repmat([0.7e1 / 0.128e3 -0.67e2 / 0.128e3 -0.55e2 / 0.192e3 0.61e2 / 0.64e2 -0.29e2 / 0.128e3 0.11e2 / 0.384e3;],[np,1]),-2:3,np,np); +Qpp(1:4,1:4)=Qpp_U; +Qpp(np-3:np,np-3:np)=flipud( fliplr(Qpp_U(1:4,1:4) ) )'; + +Qpm=-Qpp'; + +Qmp=spdiags(repmat([0.7e1 / 0.128e3 -0.67e2 / 0.128e3 -0.55e2 / 0.192e3 0.61e2 / 0.64e2 -0.29e2 / 0.128e3 0.11e2 / 0.384e3;],[nm,1]),-2:3,nm,nm); +Qmp(1:4,1:4)=Qmp_U; +Qmp(nm-3:nm,nm-3:nm)=flipud( fliplr(Qmp_U(1:4,1:4) ) )'; + +Qmm=-Qmp'; + + +Bpp=spalloc(np,np,2);Bpp(1,1)=-1;Bpp(np,np)=1; +Bmp=spalloc(nm,nm,2);Bmp(1,1)=-1;Bmp(nm,nm)=1; + +Dpp=HIp*(Qpp+1/2*Bpp) ; +Dpm=HIp*(Qpm+1/2*Bpp) ; + + +Dmp=HIm*(Qmp+1/2*Bmp) ; +Dmm=HIm*(Qmm+1/2*Bmp) ; + + +%%% Start with the staggered +Qp=spdiags(repmat([1/24 -9/8 9/8 -1/24],[np,1]),-1:2,np,nm); +Qp(1:4,1:4)=Qp_U; +Qp(np-3:np,nm-3:nm)=flipud( fliplr(-Qp_U(1:4,1:4) ) ); +Qm=-Qp'; + +Bp=spalloc(np,nm,2);Bp(1,1)=-1;Bp(np,nm)=1; +Bm=Bp'; + +Dp=HIp*(Qp+1/2*Bp) ; + +Dm=HIm*(Qm+1/2*Bm) ; + +% grids +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_upwind_6.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,78 @@ +function [xp,xm,Hp,Hm,HIp,HIm,Dp,Dm,Dpp,Dpm,Dmp,Dmm] = d1_staggered_upwind_6(n,L) + +assert(n-1 >= 12,'Not enough grid points'); + +np=n; +nm=n+1; +h = L/(n-1); + +bp=6; % Number of boundary points + + +%H- norm +Hm_U=[0.18373e5 / 0.136080e6 0 0 0 0 0; 0 0.228247e6 / 0.276480e6 0 0 0 0; 0 0 0.219557e6 / 0.207360e6 0 0 0; 0 0 0 0.44867e5 / 0.46080e5 0 0; 0 0 0 0 0.487757e6 / 0.483840e6 0; 0 0 0 0 0 0.2485447e7 / 0.2488320e7;]; +%H+ norm +Hp_U=[0.174529e6 / 0.552960e6 0 0 0 0 0; 0 0.769723e6 / 0.552960e6 0 0 0 0; 0 0 0.172613e6 / 0.276480e6 0 0 0; 0 0 0 0.343867e6 / 0.276480e6 0 0; 0 0 0 0 0.503237e6 / 0.552960e6 0; 0 0 0 0 0 0.560831e6 / 0.552960e6;]; +% Upper part of Qpp. Notice that the B matrix is not included here. +% Qpp+Qpp^T=S/2,Qpp+Qpm^T=0 +Qpp_U=[-0.11e2 / 0.12288e5 0.3248629e7 / 0.4976640e7 -0.742121e6 / 0.9953280e7 -0.173089e6 / 0.1658880e7 0.100627e6 / 0.9953280e7 0.263e3 / 0.15552e5; -0.3212989e7 / 0.4976640e7 -0.341e3 / 0.20480e5 0.488429e6 / 0.995328e6 0.2108717e7 / 0.9953280e7 0.5623e4 / 0.829440e6 -0.468779e6 / 0.9953280e7; 0.635201e6 / 0.9953280e7 -0.2135641e7 / 0.4976640e7 -0.2233e4 / 0.30720e5 0.298757e6 / 0.497664e6 -0.2148901e7 / 0.9953280e7 0.99581e5 / 0.1658880e7; 0.184969e6 / 0.1658880e7 -0.2671829e7 / 0.9953280e7 -0.1044721e7 / 0.2488320e7 -0.4697e4 / 0.30720e5 0.882599e6 / 0.995328e6 -0.2114063e7 / 0.9953280e7; -0.118447e6 / 0.9953280e7 0.15761e5 / 0.829440e6 0.915757e6 / 0.9953280e7 -0.2923243e7 / 0.4976640e7 -0.4279e4 / 0.20480e5 0.4614511e7 / 0.4976640e7; -0.263e3 / 0.15552e5 0.422447e6 / 0.9953280e7 -0.5185e4 / 0.331776e6 0.424727e6 / 0.9953280e7 -0.570779e6 / 0.995328e6 -0.2761e4 / 0.12288e5;]; + +% Upper part of Qmp. Notice that the B matrix is not included here. +% Qmp+Qmp^T=S/2,Qmp+Qmm^T=0 +Qmp_U=[-0.6660404399e10 / 0.975680535935e12 0.42802131970831759e17 / 0.60695134779444480e17 -0.27241603626152813e17 / 0.91042702169166720e17 0.148772145985039e15 / 0.1264481974571760e16 -0.386865438537449e15 / 0.30347567389722240e17 -0.739491571084877e15 / 0.182085404338333440e18; -0.40937739835891039e17 / 0.60695134779444480e17 -0.20934998251893e14 / 0.570912496455680e15 0.3069347264824655e16 / 0.3082927480860672e16 -0.531249064089227e15 / 0.1541463740430336e16 0.6343721417240047e16 / 0.107902461830123520e18 0.194756943144697e15 / 0.138731736638730240e18; 0.24212381292051773e17 / 0.91042702169166720e17 -0.13932096926615587e17 / 0.15414637404303360e17 -0.22149140293797e14 / 0.285456248227840e15 0.443549615179363e15 / 0.481707418884480e15 -0.1531771257444041e16 / 0.5994581212784640e16 0.23579222779798361e17 / 0.416195209916190720e18; -0.119651391006031e15 / 0.1264481974571760e16 0.2061363806050549e16 / 0.7707318702151680e16 -0.355212104634871e15 / 0.481707418884480e15 -0.42909037900311e14 / 0.285456248227840e15 0.25466291778687943e17 / 0.26975615457530880e17 -0.3289076301679109e16 / 0.11560978053227520e17; 0.153994229603129e15 / 0.30347567389722240e17 -0.2655886084488631e16 / 0.107902461830123520e18 0.782300684927837e15 / 0.5994581212784640e16 -0.17511430871269903e17 / 0.26975615457530880e17 -0.413193098349471e15 / 0.1998193737594880e16 0.189367309285289755e18 / 0.194224431294222336e18; 0.894580992211517e15 / 0.182085404338333440e18 -0.209441083772219e15 / 0.27746347327746048e17 -0.4946149632449393e16 / 0.416195209916190720e18 0.334964642443661e15 / 0.2890244513306880e16 -0.604469352802317407e18 / 0.971122156471111680e18 -0.128172128502407e15 / 0.570912496455680e15;]; + +% The staggered + operator, upper part. Notice that the B matrix is not included here.Qp+Qm^T=0 +Qp_U=[-0.34660470729017653729e20 / 0.113641961250214656000e21 0.351671379135966469961e21 / 0.415604886857927884800e21 0.1819680091728191503e19 / 0.103901221714481971200e21 -0.18252344147469061739e20 / 0.346337405714939904000e21 -0.18145368485798816351e20 / 0.727308552001373798400e21 0.2627410615589536403e19 / 0.138534962285975961600e21; -0.1606450873889019037e19 / 0.7576130750014310400e19 -0.8503979509850519441e19 / 0.9235664152398397440e19 0.2208731907526094393e19 / 0.2308916038099599360e19 0.1143962309827873891e19 / 0.7696386793665331200e19 0.1263616990270014071e19 / 0.16162412266697195520e20 -0.1402288892096389187e19 / 0.27706992457195192320e20; -0.502728075208147729e18 / 0.11364196125021465600e20 0.5831273443201206481e19 / 0.41560488685792788480e20 -0.9031420599281409001e19 / 0.10390122171448197120e20 0.29977986617775158621e20 / 0.34633740571493990400e20 -0.7995649008389734663e19 / 0.72730855200137379840e20 0.728315692435313537e18 / 0.41560488685792788480e20; 0.710308100786581369e18 / 0.11364196125021465600e20 -0.2317346723533341809e19 / 0.41560488685792788480e20 -0.1357359577229545879e19 / 0.10390122171448197120e20 -0.35124499190079631261e20 / 0.34633740571493990400e20 0.81675241511291974823e20 / 0.72730855200137379840e20 0.144034831596315317e18 / 0.13853496228597596160e20; 0.13360631165154733e17 / 0.841792305557145600e18 -0.875389186128426797e18 / 0.27706992457195192320e20 0.95493318392786453e17 / 0.6926748114298798080e19 0.1714625642820850967e19 / 0.23089160380995993600e20 -0.53371483072841696197e20 / 0.48487236800091586560e20 0.30168063964639488547e20 / 0.27706992457195192320e20; -0.1943232250834614071e19 / 0.113641961250214656000e21 0.8999269402554660119e19 / 0.415604886857927884800e21 0.1242786058815312817e19 / 0.103901221714481971200e21 -0.7480218714053301461e19 / 0.346337405714939904000e21 0.28468183824757664191e20 / 0.727308552001373798400e21 -0.476082521721490837529e21 / 0.415604886857927884800e21;]; + +Hp=spdiags(ones(np,1),0,np,np); +Hp(1:bp,1:bp)=Hp_U; +Hp(np-bp+1:np,np-bp+1:np)=fliplr(flipud(Hp_U)); +Hp=Hp*h; +HIp=inv(Hp); + +Hm=spdiags(ones(nm,1),0,nm,nm); +Hm(1:bp,1:bp)=Hm_U; +Hm(nm-bp+1:nm,nm-bp+1:nm)=fliplr(flipud(Hm_U)); +Hm=Hm*h; +HIm=inv(Hm); + +Qpp=spdiags(repmat([-0.157e3 / 0.15360e5 0.537e3 / 0.5120e4 -0.3147e4 / 0.5120e4 -0.231e3 / 0.1024e4 0.999e3 / 0.1024e4 -0.1461e4 / 0.5120e4 0.949e3 / 0.15360e5 -0.33e2 / 0.5120e4;],[np,1]),-3:4,np,np); +Qpp(1:bp,1:bp)=Qpp_U; +Qpp(np-bp+1:np,np-bp+1:np)=flipud( fliplr(Qpp_U ) )'; + +Qpm=-Qpp'; + +Qmp=spdiags(repmat([-0.157e3 / 0.15360e5 0.537e3 / 0.5120e4 -0.3147e4 / 0.5120e4 -0.231e3 / 0.1024e4 0.999e3 / 0.1024e4 -0.1461e4 / 0.5120e4 0.949e3 / 0.15360e5 -0.33e2 / 0.5120e4;],[nm,1]),-3:4,nm,nm); +Qmp(1:bp,1:bp)=Qmp_U; +Qmp(nm-bp+1:nm,nm-bp+1:nm)=flipud( fliplr(Qmp_U ) )'; + +Qmm=-Qmp'; + +Bpp=spalloc(np,np,2);Bpp(1,1)=-1;Bpp(np,np)=1; +Bmp=spalloc(nm,nm,2);Bmp(1,1)=-1;Bmp(nm,nm)=1; + + +Dpp=HIp*(Qpp+1/2*Bpp) ; +Dpm=HIp*(Qpm+1/2*Bpp) ; + + +Dmp=HIm*(Qmp+1/2*Bmp) ; +Dmm=HIm*(Qmm+1/2*Bmp) ; + + +%%% Start with the staggered +Qp=spdiags(repmat([-0.3e1 / 0.640e3 0.25e2 / 0.384e3 -0.75e2 / 0.64e2 0.75e2 / 0.64e2 -0.25e2 / 0.384e3 0.3e1 / 0.640e3],[np,1]),-2:3,np,nm); +Qp(1:bp,1:bp)=Qp_U; +Qp(np-bp+1:np,nm-bp+1:nm)=flipud( fliplr(-Qp_U ) ); +Qm=-Qp'; + +Bp=spalloc(np,nm,2);Bp(1,1)=-1;Bp(np,nm)=1; +Bm=Bp'; + +Dp=HIp*(Qp+1/2*Bp) ; + +Dm=HIm*(Qm+1/2*Bm) ; + +% grids +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/+implementations/d1_staggered_upwind_8.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,82 @@ +function [xp,xm,Hp,Hm,HIp,HIm,Dp,Dm,Dpp,Dpm,Dmp,Dmm] = d1_staggered_upwind_8(n, L) + +assert(n-1 >= 12,'Not enough grid points'); + +np=n; +nm=n+1; +h = L/(n-1); + +bp=8; % Number of boundary points + + +%H- norm +Hm_U=[0.27058177e8 / 0.194594400e9 0 0 0 0 0 0 0; 0 0.378196601e9 / 0.464486400e9 0 0 0 0 0 0; 0 0 0.250683799e9 / 0.232243200e9 0 0 0 0 0; 0 0 0 0.48820787e8 / 0.51609600e8 0 0 0 0; 0 0 0 0 0.23953873e8 / 0.23224320e8 0 0 0; 0 0 0 0 0 0.55018021e8 / 0.55738368e8 0 0; 0 0 0 0 0 0 0.284772253e9 / 0.283852800e9 0; 0 0 0 0 0 0 0 0.6036097241e10 / 0.6038323200e10;]; + +%H+ norm +Hp_U=[0.273925927e9 / 0.928972800e9 0 0 0 0 0 0 0; 0 0.1417489391e10 / 0.928972800e9 0 0 0 0 0 0; 0 0 0.26528603e8 / 0.103219200e9 0 0 0 0 0; 0 0 0 0.66843235e8 / 0.37158912e8 0 0 0 0; 0 0 0 0 0.76542481e8 / 0.185794560e9 0 0 0; 0 0 0 0 0 0.132009637e9 / 0.103219200e9 0 0; 0 0 0 0 0 0 0.857580049e9 / 0.928972800e9 0; 0 0 0 0 0 0 0 0.937663193e9 / 0.928972800e9;]; + +% Upper part of Qpp. Notice that the B matrix is not included here. +% Qpp+Qpp^T=S/2,Qpp+Qpm^T=0 +Qpp_U=[-0.5053e4 / 0.43253760e8 0.299126491231e12 / 0.442810368000e12 -0.562745123e9 / 0.8856207360e10 -0.59616938177e11 / 0.398529331200e12 -0.5489464381e10 / 0.177124147200e12 0.6994759151e10 / 0.88562073600e11 0.712609553e9 / 0.181149696000e12 -0.14167e5 / 0.998400e6; -0.74652297589e11 / 0.110702592000e12 -0.995441e6 / 0.302776320e9 0.662285893e9 / 0.2236416000e10 0.2364992837e10 / 0.5535129600e10 0.8893339297e10 / 0.49816166400e11 -0.2166314077e10 / 0.8945664000e10 -0.78813191e8 / 0.2952069120e10 0.88810243397e11 / 0.1992646656000e13; 0.16939159e8 / 0.276756480e9 -0.2302477691e10 / 0.8200192000e10 -0.197067e6 / 0.9175040e7 0.46740527413e11 / 0.88562073600e11 -0.4827794723e10 / 0.11070259200e11 0.155404199e9 / 0.1230028800e10 0.15110006581e11 / 0.295206912000e12 -0.2491856531e10 / 0.88562073600e11; 0.7568509969e10 / 0.49816166400e11 -0.19762404121e11 / 0.44281036800e11 -0.2554553593e10 / 0.5535129600e10 -0.19550057e8 / 0.302776320e9 0.145135201e9 / 0.210862080e9 0.1596117533e10 / 0.11070259200e11 0.104983477e9 / 0.3558297600e10 -0.1087937837e10 / 0.25303449600e11; 0.5282544031e10 / 0.177124147200e12 -0.131784830977e12 / 0.797058662400e12 0.8310607171e10 / 0.22140518400e11 -0.56309573e8 / 0.105431040e9 -0.36477607e8 / 0.302776320e9 0.57443289107e11 / 0.88562073600e11 -0.220576549e9 / 0.691891200e9 0.39942520697e11 / 0.398529331200e12; -0.1743516779e10 / 0.22140518400e11 0.7784403199e10 / 0.32800768000e11 -0.459036521e9 / 0.4920115200e10 -0.5749179391e10 / 0.22140518400e11 -0.562460513e9 / 0.1383782400e10 -0.1500741e7 / 0.9175040e7 0.70986504791e11 / 0.73801728000e11 -0.930160589e9 / 0.3406233600e10; -0.712609553e9 / 0.181149696000e12 0.180761e6 / 0.6589440e7 -0.18016744831e11 / 0.295206912000e12 0.18651112477e11 / 0.797058662400e12 0.7155507361e10 / 0.44281036800e11 -0.49358401541e11 / 0.73801728000e11 -0.55032223e8 / 0.302776320e9 0.38275518139e11 / 0.40255488000e11; 0.14167e5 / 0.998400e6 -0.88810243397e11 / 0.1992646656000e13 0.650307179e9 / 0.22140518400e11 0.474654619e9 / 0.16102195200e11 -0.7267828861e10 / 0.199264665600e12 0.42227833e8 / 0.425779200e9 -0.71245142351e11 / 0.110702592000e12 -0.7998899e7 / 0.43253760e8;]; + +% Upper part of Qmp. Notice that the B matrix is not included here. +% Qmp+Qmp^T=S/2,Qmp+Qmm^T=0 +Qmp_U=[-0.92025012754706822244637e23 / 0.73350939131274317328275670e26 0.930302337374620084855601123690977e33 / 0.1359398284732080668181467336976000e34 -0.727851704787797291000113057457371e33 / 0.2718796569464161336362934673952000e34 0.7360201789777281105403766444579e31 / 0.90626552315472044545431155798400e32 0.13716157770579047943179985700303e32 / 0.543759313892832267272586934790400e33 -0.553550686983724599329589830113e30 / 0.21750372555713290690903477391616e32 0.39889728754596585193681401053e29 / 0.20596943708061828305779808136000e32 0.4597826214543803453588826224861e31 / 0.2718796569464161336362934673952000e34; -0.460817194517403953445488602736801e33 / 0.679699142366040334090733668488000e33 -0.2251237342833501172927156567e28 / 0.267062619272621870023659683840e30 0.190175037040902421814031988319053e33 / 0.202800676510147232549216572416000e33 -0.280264041304585424221475951435491e33 / 0.1081603608054118573595821719552000e34 -0.2948540748840639854083293613843e31 / 0.81120270604058893019686628966400e32 0.10373234521893519136055741251159e32 / 0.194688649449741343247247909519360e33 -0.391946441371315598560753809131e30 / 0.59488198442976521547770194575360e32 -0.15471439859462263133402977429037e32 / 0.6026077244872946338605292437504000e34; 0.702728186606917922841148808013121e33 / 0.2718796569464161336362934673952000e34 -0.738964213741742941634289388463587e33 / 0.811202706040588930196866289664000e33 -0.6938440904692701484464706797e28 / 0.267062619272621870023659683840e30 0.13587935221144065448123508546711e32 / 0.16900056375845602712434714368000e32 -0.398581023027193812136121833051e30 / 0.3244810824162355720787465158656e31 -0.4390487184318690815376655433561e31 / 0.486721623624353358118119773798400e33 0.39375020562052974775725982794643e32 / 0.5948819844297652154777019457536000e34 -0.337431206994896862016299739723e30 / 0.1054563517852765609255926176563200e34; -0.1626484714285090813572964936931e31 / 0.22656638078868011136357788949600e32 0.247081446636583523913555235517491e33 / 0.1081603608054118573595821719552000e34 -0.98606689532786656855690277893813e32 / 0.135200451006764821699477714944000e33 -0.54412910204947725598668603049e29 / 0.801187857817865610070979051520e30 0.44527159063104584858762191285733e32 / 0.54080180402705928679791085977600e32 -0.4170461618635408764831166558231e31 / 0.18541776138070604118785515192320e32 0.2733053052508781643445069010989e31 / 0.55081665224978260692379809792000e32 -0.62697761224771731704532896739409e32 / 0.7030423452351770728372841177088000e34; -0.16732447706243527499842078942603e32 / 0.543759313892832267272586934790400e33 0.166114324967929244010272382781e30 / 0.2897152521573531893560236748800e31 0.45101984053799299547057123965e29 / 0.811202706040588930196866289664e30 -0.17968772701532140224971787578029e32 / 0.27040090201352964339895542988800e32 -0.48635228792591105226576208151e29 / 0.400593928908932805035489525760e30 0.89480251714879473157102672858403e32 / 0.97344324724870671623623954759680e32 -0.42143863277048975845235502012773e32 / 0.148720496107441303869425486438400e33 0.4389611982769118812600028148913e31 / 0.52728175892638280462796308828160e32; 0.590475930631333482244268674627e30 / 0.21750372555713290690903477391616e32 -0.1702402353852012839500776925027e31 / 0.27812664207105906178178272788480e32 0.5538920825569170589993365115709e31 / 0.121680405906088339529529943449600e33 0.13866428236712167588433125984777e32 / 0.129792432966494228831498606346240e33 -0.16470253483258635888811378530287e32 / 0.24336081181217667905905988689920e32 -0.391812951622286986994545412081e30 / 0.2403563573453596830212937154560e31 0.345131812155051195787190432571781e33 / 0.356929190657859129286621167452160e33 -0.8126635967231661246920267342728147e34 / 0.25309524428466374622142228237516800e35; -0.88170259036818455465427409231e29 / 0.41193887416123656611559616272000e32 0.943459254925453305858100463659e30 / 0.118976396885953043095540389150720e33 -0.104249970715879362499981962934393e33 / 0.5948819844297652154777019457536000e34 0.78160374875076232344526852587e29 / 0.18360555074992753564126603264000e32 0.37536035579237342566706229296521e32 / 0.297440992214882607738850972876800e33 -0.240812609144054591374890622842541e33 / 0.356929190657859129286621167452160e33 -0.145390363516271219220036634153e30 / 0.801187857817865610070979051520e30 0.38093040928245760105862644889779897e35 / 0.38667328987934739006050626473984000e35; -0.4596580943680048141920105029111e31 / 0.2718796569464161336362934673952000e34 0.15262190991038620305994595494037e32 / 0.6026077244872946338605292437504000e34 0.1782508142749448850000523441843e31 / 0.1054563517852765609255926176563200e34 -0.33510338065544202178903991034341e32 / 0.7030423452351770728372841177088000e34 -0.8226959851063633384855147175189e31 / 0.421825407141106243702370470625280e33 0.3729744974003821244717412110773747e34 / 0.25309524428466374622142228237516800e35 -0.6554525408845613610278033864711443e34 / 0.9666832246983684751512656618496000e34 -0.49383212966905764275823531781e29 / 0.267062619272621870023659683840e30;]; + +% The staggered + operator, upper part. Notice that the B matrix is not included here.Qp+Qm^T=0 +Qp_U=[-0.6661444046602902130086192779621746771e37 / 0.23342839720855518078613888837079040000e38 0.12367666586033683088530161144944922123649e41 / 0.14797137255429936031548004199961722880000e41 0.4395151478839780503265910147432452357e37 / 0.186518536833150454179176523528929280000e39 -0.87963490365651280757669626389462038003e38 / 0.1345194295948176002868000381814702080000e40 -0.224187853266917333808369723332131873619e39 / 0.5178998039400477611041801469986603008000e40 0.55314264508663245255306547445101748003e38 / 0.2959427451085987206309600839992344576000e40 0.232303691019191510116997121080032335473e39 / 0.7398568627714968015774002099980861440000e40 -0.2816079756494683118054172509551114287e37 / 0.182680706857159704093185237036564480000e39; -0.10353150194859080046224306922028068797e38 / 0.43350988053017390717425793554575360000e38 -0.41810120772068966379630018024826564868789e41 / 0.44391411766289808094644012599885168640000e41 0.14548989645937048285445132077600344947e38 / 0.16118885899161150361163403267932160000e38 0.1353060140543056426043612771323929599939e40 / 0.6341630252327115442092001799983595520000e40 0.52584560489710238297754746193308329991e38 / 0.317081512616355772104600089999179776000e39 -0.322880558646947920788817445027309369183e39 / 0.8878282353257961618928802519977033728000e40 -0.2523050363123545986203815545725359199053e40 / 0.22195705883144904047322006299942584320000e41 0.2171073153815036601208579021528549178587e40 / 0.44391411766289808094644012599885168640000e41; -0.891108828407068615176254357157902263e36 / 0.14450329351005796905808597851525120000e38 0.9207480733237809022786174965361787199767e40 / 0.44391411766289808094644012599885168640000e41 -0.42541158717297365539935073107511004261e38 / 0.62172845611050151393058841176309760000e38 0.3752589973871193536179209598104238527889e40 / 0.4932379085143312010516001399987240960000e40 -0.503978909830206767329311700636686423011e39 / 0.2219570588314490404732200629994258432000e40 -0.25729240076595732139825342259542295873e38 / 0.328825272342887467367733426665816064000e39 0.283553624900223723890865927423934504751e39 / 0.2466189542571656005258000699993620480000e40 -0.129077863120107719239717930927898530811e39 / 0.4035582887844528008604001145444106240000e40; 0.150348887575956035991597139943595613e36 / 0.2000814833216187263881190471749632000e37 -0.20271017027971837857434511014725441049e38 / 0.422775350155141029472800119998906368000e39 -0.28593401740189393456434271848012721991e38 / 0.87041983855470211950282377646833664000e38 -0.3107901953269564253446821657614993689969e40 / 0.2959427451085987206309600839992344576000e40 0.163990240717967340033770840655582198979e39 / 0.147971372554299360315480041999617228800e39 0.1282390462809880153203810439898877805771e40 / 0.5326969411954776971357281511986220236800e40 0.55264142141446425784998699697993554089e38 / 0.1479713725542993603154800419996172288000e40 -0.11464469241412665723941542481946601159e38 / 0.328825272342887467367733426665816064000e39; 0.523088046329055388999216632374482217e36 / 0.8670197610603478143485158710915072000e37 -0.1060078188350823496391913491020744512571e40 / 0.8878282353257961618928802519977033728000e40 0.2489758378222482566046953325470732791e37 / 0.87041983855470211950282377646833664000e38 0.2223322627542429729644405110485179851147e40 / 0.8878282353257961618928802519977033728000e40 -0.405229856535104846314030690968355556777e39 / 0.443914117662898080946440125998851686400e39 0.1490365162783652350291746570529147907183e40 / 0.1775656470651592323785760503995406745600e40 -0.768972334347761129534143069144613163467e39 / 0.4439141176628980809464401259988516864000e40 0.34866718651627637034229483614924933299e38 / 0.1268326050465423088418400359996719104000e40; -0.522920230014765612739540250860177277e36 / 0.14450329351005796905808597851525120000e38 0.1453599225900263767517831305438030179313e40 / 0.44391411766289808094644012599885168640000e41 0.39466073050115777575909309244731209107e38 / 0.435209919277351059751411888234168320000e39 -0.501532616891797921922210650058733239369e39 / 0.4932379085143312010516001399987240960000e40 -0.276688880738967796756224778348832938429e39 / 0.2219570588314490404732200629994258432000e40 -0.346414321340583244146983258114429082007e39 / 0.328825272342887467367733426665816064000e39 0.12835585968364300662881260070481636919e38 / 0.10676145205937904784666669696942080000e38 -0.823464330534329517579044232163467356639e39 / 0.44391411766289808094644012599885168640000e41; -0.52802811745049309885720368328150617e35 / 0.1688999534533145092886719229399040000e37 0.306081878756402097710283710502974342821e39 / 0.4932379085143312010516001399987240960000e40 -0.43281676756908448328630499995327205907e38 / 0.1305629757832053179254235664702504960000e40 -0.136777517026276610492513752852976312597e39 / 0.4932379085143312010516001399987240960000e40 0.7290267185112688983569547959866321207e37 / 0.246618954257165600525800069999362048000e39 0.351639553804386998900032061955116226307e39 / 0.3804978151396269265255201079990157312000e40 -0.2840607921448362125113163437018746783083e40 / 0.2466189542571656005258000699993620480000e40 0.1859197240206073478058984196606793676119e40 / 0.1644126361714437336838667133329080320000e40; 0.5412884950805861225701675068383948563e37 / 0.303456916371121735021980554882027520000e39 -0.1279848124286614098666833963684894093627e40 / 0.44391411766289808094644012599885168640000e41 0.17218770500911534891873213313315117e35 / 0.39564538116122823613764717112197120000e38 0.904771800018341402297740826650985365019e39 / 0.44391411766289808094644012599885168640000e41 0.65377500469171784914136353007658339737e38 / 0.15536994118201432833125404409959809024000e41 -0.211014619053462658139013021424372225649e39 / 0.8878282353257961618928802519977033728000e40 0.195325945159072852492812627815477159003e39 / 0.3170815126163557721046000899991797760000e40 -0.52260858238454846311625894178508086247499e41 / 0.44391411766289808094644012599885168640000e41;]; + +Hp=spdiags(ones(np,1),0,np,np); +Hp(1:bp,1:bp)=Hp_U; +Hp(np-bp+1:np,np-bp+1:np)=fliplr(flipud(Hp_U)); +Hp=Hp*h; +HIp=inv(Hp); + +Hm=spdiags(ones(nm,1),0,nm,nm); +Hm(1:bp,1:bp)=Hm_U; +Hm(nm-bp+1:nm,nm-bp+1:nm)=fliplr(flipud(Hm_U)); +Hm=Hm*h; +HIm=inv(Hm); + +tt=[0.1447e4 / 0.688128e6 -0.17119e5 / 0.688128e6 0.8437e4 / 0.57344e5 -0.5543e4 / 0.8192e4 -0.15159e5 / 0.81920e5 0.16139e5 / 0.16384e5 -0.2649e4 / 0.8192e4 0.15649e5 / 0.172032e6 -0.3851e4 / 0.229376e6 0.5053e4 / 0.3440640e7;]; + +Qpp=spdiags(repmat(tt,[np,1]),-4:5,np,np); +Qpp(1:bp,1:bp)=Qpp_U; +Qpp(np-bp+1:np,np-bp+1:np)=flipud( fliplr(Qpp_U ) )'; + +Qpm=-Qpp'; + +Qmp=spdiags(repmat(tt,[nm,1]),-4:5,nm,nm); +Qmp(1:bp,1:bp)=Qmp_U; +Qmp(nm-bp+1:nm,nm-bp+1:nm)=flipud( fliplr(Qmp_U ) )'; + +Qmm=-Qmp'; + + +Bpp=spalloc(np,np,2);Bpp(1,1)=-1;Bpp(np,np)=1; +Bmp=spalloc(nm,nm,2);Bmp(1,1)=-1;Bmp(nm,nm)=1; + +Dpp=HIp*(Qpp+1/2*Bpp) ; +Dpm=HIp*(Qpm+1/2*Bpp) ; + + +Dmp=HIm*(Qmp+1/2*Bmp) ; +Dmm=HIm*(Qmm+1/2*Bmp) ; + + +%%% Start with the staggered +Qp=spdiags(repmat([0.5e1 / 0.7168e4 -0.49e2 / 0.5120e4 0.245e3 / 0.3072e4 -0.1225e4 / 0.1024e4 0.1225e4 / 0.1024e4 -0.245e3 / 0.3072e4 0.49e2 / 0.5120e4 -0.5e1 / 0.7168e4;],[np,1]),-3:4,np,nm); +Qp(1:bp,1:bp)=Qp_U; +Qp(np-bp+1:np,nm-bp+1:nm)=flipud( fliplr(-Qp_U ) ); +Qm=-Qp'; + +Bp=spalloc(np,nm,2);Bp(1,1)=-1;Bp(np,nm)=1; +Bm=Bp'; + +Dp=HIp*(Qp+1/2*Bp) ; + +Dm=HIm*(Qm+1/2*Bm) ; + +% grids +xp = h*[0:n]'; +xm = h*[0 1/2+0:n n]'; \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/D1Staggered.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,86 @@ +classdef D1Staggered < sbp.OpSet + properties + % x_primal: "primal" grid with m points. Equidistant. Called Plus grid in Ossian's paper. + % x_dual: "dual" grid with m+1 points. Called Minus grid in Ossian's paper. + + % D1_primal takes FROM dual grid TO primal grid + % D1_dual takes FROM primal grid TO dual grid + + D1_primal % SBP operator approximating first derivative + D1_dual % SBP operator approximating first derivative + H_primal % Norm matrix + H_dual % Norm matrix + H_primalI % H^-1 + H_dualI % H^-1 + e_primal_l % Left boundary operator + e_dual_l % Left boundary operator + e_primal_r % Right boundary operator + e_dual_r % Right boundary operator + m % Number of grid points. + m_primal % Number of grid points. + m_dual % Number of grid points. + h % Step size + x_primal % grid + x_dual % grid + x + borrowing % Struct with borrowing limits for different norm matrices + end + + methods + function obj = D1Staggered(m,lim,order) + + x_l = lim{1}; + x_r = lim{2}; + L = x_r-x_l; + + m_primal = m; + m_dual = m+1; + + switch order + case 2 + [x_primal, x_dual, Pp, Pm, Qp, Qm] = sbp.implementations.d1_staggered_2(m, L); + case 4 + [x_primal, x_dual, Pp, Pm, Qp, Qm] = sbp.implementations.d1_staggered_4(m, L); + case 6 + [x_primal, x_dual, Pp, Pm, Qp, Qm] = sbp.implementations.d1_staggered_6(m, L); + otherwise + error('Invalid operator order %d.',order); + end + + obj.m = m; + obj.m_primal = m_primal; + obj.m_dual = m_dual; + obj.x_primal = x_l + x_primal'; + obj.x_dual = x_l + x_dual'; + + D1_primal = Pp\Qp; + D1_dual = Pm\Qm; + + obj.D1_primal = D1_primal; + obj.D1_dual = D1_dual; + obj.H_primal = Pp; + obj.H_dual = Pm; + + obj.e_primal_l = sparse(m_primal,1); + obj.e_primal_r = sparse(m_primal,1); + obj.e_primal_l(1) = 1; + obj.e_primal_r(m_primal) = 1; + + obj.e_dual_l = sparse(m_dual,1); + obj.e_dual_r = sparse(m_dual,1); + obj.e_dual_l(1) = 1; + obj.e_dual_r(m_dual) = 1; + + obj.H_primalI = inv(obj.H_primal); + obj.H_dualI = inv(obj.H_dual); + + obj.borrowing = []; + obj.x = []; + + end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/D1StaggeredUpwind.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,95 @@ +classdef D1StaggeredUpwind < sbp.OpSet + % Compatible staggered and upwind operators by Ken Mattsson and Ossian O'reilly + properties + % x_primal: "primal" grid with m points. Equidistant. Called Plus grid in Ossian's paper. + % x_dual: "dual" grid with m+1 points. Called Minus grid in Ossian's paper. + + % D1_primal takes FROM dual grid TO primal grid + % D1_dual takes FROM primal grid TO dual grid + + D1_primal % SBP operator approximating first derivative + D1_dual % SBP operator approximating first derivative + + Dplus_primal % Upwind operator on primal grid + Dminus_primal % Upwind operator on primal grid + Dplus_dual % Upwind operator on dual grid + Dminus_dual % Upwind operator on dual grid + + H_primal % Norm matrix + H_dual % Norm matrix + H_primalI % H^-1 + H_dualI % H^-1 + e_primal_l % Left boundary operator + e_dual_l % Left boundary operator + e_primal_r % Right boundary operator + e_dual_r % Right boundary operator + m % Number of grid points. + m_primal % Number of grid points. + m_dual % Number of grid points. + h % Step size + x_primal % grid + x_dual % grid + x + borrowing % Struct with borrowing limits for different norm matrices + end + + methods + function obj = D1StaggeredUpwind(m,lim,order) + + x_l = lim{1}; + x_r = lim{2}; + L = x_r-x_l; + + m_primal = m; + m_dual = m+1; + + switch order + case 2 + [obj.x_primal, obj.x_dual, obj.H_primal, obj.H_dual,... + obj.H_primalI, obj.H_dualI,... + obj.D1_primal, obj.D1_dual, obj.Dplus_primal, obj.Dminus_primal,... + obj.Dplus_dual, obj.Dminus_dual] = sbp.implementations.d1_staggered_upwind_2(m, L); + case 4 + [obj.x_primal, obj.x_dual, obj.H_primal, obj.H_dual,... + obj.H_primalI, obj.H_dualI,... + obj.D1_primal, obj.D1_dual, obj.Dplus_primal, obj.Dminus_primal,... + obj.Dplus_dual, obj.Dminus_dual] = sbp.implementations.d1_staggered_upwind_4(m, L); + case 6 + [obj.x_primal, obj.x_dual, obj.H_primal, obj.H_dual,... + obj.H_primalI, obj.H_dualI,... + obj.D1_primal, obj.D1_dual, obj.Dplus_primal, obj.Dminus_primal,... + obj.Dplus_dual, obj.Dminus_dual] = sbp.implementations.d1_staggered_upwind_6(m, L); + case 8 + [obj.x_primal, obj.x_dual, obj.H_primal, obj.H_dual,... + obj.H_primalI, obj.H_dualI,... + obj.D1_primal, obj.D1_dual, obj.Dplus_primal, obj.Dminus_primal,... + obj.Dplus_dual, obj.Dminus_dual] = sbp.implementations.d1_staggered_upwind_8(m, L); + otherwise + error('Invalid operator order %d.',order); + end + + obj.m = m; + obj.m_primal = m_primal; + obj.m_dual = m_dual; + obj.h = L/(m-1); + + obj.e_primal_l = sparse(m_primal,1); + obj.e_primal_r = sparse(m_primal,1); + obj.e_primal_l(1) = 1; + obj.e_primal_r(m_primal) = 1; + + obj.e_dual_l = sparse(m_dual,1); + obj.e_dual_r = sparse(m_dual,1); + obj.e_dual_l(1) = 1; + obj.e_dual_r(m_dual) = 1; + + obj.borrowing = []; + obj.x = []; + + end + + function str = string(obj) + str = [class(obj) '_' num2str(obj.order)]; + end + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+scheme/Staggered1DAcoustics.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,231 @@ +classdef Staggered1DAcoustics < scheme.Scheme + properties + m % Number of points of primal grid in each direction, possibly a vector + h % Grid spacing + + % Grids + grid % Total grid object + grid_primal + grid_dual + + order % Order accuracy for the approximation + + H % Combined norm + Hi % Inverse + D % Semi-discrete approximation matrix + + D1_primal + D1_dual + + % Pick out left or right boundary + e_l + e_r + + % Initial data + v0 + + % Pick out primal or dual component + e_primal + e_dual + + % System matrices + A + B + end + + + methods + % Scheme for A*u_t + B u_x = 0, + % u = [p, v]; + % A: Diagonal and A > 0, + % A = [a1, 0; + % 0, a2] + % + % B = B^T and with diagonal entries = 0. + % B = [0 b + % b 0] + % Here we store p on the primal grid and v on the dual + function obj = Staggered1DAcoustics(g, order, A, B) + default_arg('A',[1, 0; 0, 1]); + default_arg('B',[0, 1; 1, 0]); + + obj.order = order; + obj.A = A; + obj.B = B; + + % Grids + obj.m = g.size(); + xl = g.getBoundary('l'); + xr = g.getBoundary('r'); + xlim = {xl, xr}; + + obj.grid = g; + obj.grid_primal = g.grids{1}; + obj.grid_dual = g.grids{2}; + + % Get operators + ops = sbp.D1StaggeredUpwind(obj.m, xlim, order); + obj.h = ops.h; + + % Build combined operators + H_primal = ops.H_primal; + H_dual = ops.H_dual; + obj.H = blockmatrix.toMatrix( {H_primal, []; [], H_dual } ); + obj.Hi = inv(obj.H); + + D1_primal = ops.D1_primal; + D1_dual = ops.D1_dual; + D = {[], -1/A(1,1)*B(1,2)*D1_primal;... + -1/A(2,2)*B(2,1)*D1_dual, []}; + obj.D = blockmatrix.toMatrix(D); + obj.D1_primal = D1_primal; + obj.D1_dual = D1_dual; + + % Combined boundary operators + e_primal_l = ops.e_primal_l; + e_primal_r = ops.e_primal_r; + e_dual_l = ops.e_dual_l; + e_dual_r = ops.e_dual_r; + e_l = {e_primal_l, [];... + [] , e_dual_l}; + e_r = {e_primal_r, [];... + [] , e_dual_r}; + obj.e_l = blockmatrix.toMatrix(e_l); + obj.e_r = blockmatrix.toMatrix(e_r); + + % Pick out first or second component of solution + N_primal = obj.grid_primal.N(); + N_dual = obj.grid_dual.N(); + obj.e_primal = [speye(N_primal, N_primal); sparse(N_dual, N_primal)]; + obj.e_dual = [sparse(N_primal, N_dual); speye(N_dual, N_dual)]; + + + end + % Closure functions return the operators applied to the own domain to close the boundary + % Penalty functions return the operators to force the solution. In the case of an interface it returns the operator applied to the other domain. + % 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','p'); + + % type = 'p' => boundary condition for p + % type = 'v' => boundary condition for v + % type = 'characteristic' => incoming characteristc = data + % No other types implemented yet + + % BC on the form Lu - g = 0; + + % Need a transformation T such that w = T^{−1}*u is a + % meaningful change of variables and T^T*B*T is block-diagonal, + % For linear acoustics, T = T_C meets these criteria. + + A = obj.A; + B = obj.B; + C = inv(A)*B; + + % Diagonalize C and use T_C to diagonalize B + [T, ~] = eig(C); + Lambda = T'*B*T; + lambda = diag(Lambda); + + + % Identify in- and outgoing variables + Iplus = lambda > 0; + Iminus = lambda < 0; + + switch boundary + case 'l' + Iin = Iplus; + case 'r' + Iin = Iminus; + end + Tin = T(:,Iin); + + switch type + case 'p' + L = [1, 0]; + case 'v' + L = [0, 1]; + case 'characteristic' + % Diagonalize C + [T_C, Lambda_C] = eig(C); + lambda_C = diag(Lambda_C); + % Identify in- and outgoing characteristic variables + Iplus = lambda_C > 0; + Iminus = lambda_C < 0; + + switch boundary + case 'l' + Iin_C = Iplus; + case 'r' + Iin_C = Iminus; + end + T_C_inv = inv(T_C); + L = T_C_inv(Iin_C,:); + otherwise + error('Boundary condition not implemented.'); + end + + % Penalty parameters + sigma = [0; 0]; + sigma(Iin) = lambda(Iin); + + % Sparsify + A = sparse(A); + T = sparse(T); + sigma = sparse(sigma); + L = sparse(L); + Tin = sparse(Tin); + + switch boundary + case 'l' + tau = -1*obj.e_l * inv(A) * inv(T)' * sigma * inv(L*Tin); + closure = obj.Hi*tau*L*obj.e_l'; + + case 'r' + tau = 1*obj.e_r * inv(A) * inv(T)' * sigma * inv(L*Tin); + closure = obj.Hi*tau*L*obj.e_r'; + + end + + penalty = -obj.Hi*tau; + + end + + function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + + error('Staggered1DAcoustics, interface not implemented'); + + switch boundary + % Upwind coupling + case {'l','left'} + tau = -1*obj.e_l; + closure = obj.Hi*tau*obj.e_l'; + penalty = -obj.Hi*tau*neighbour_scheme.e_r'; + case {'r','right'} + tau = 0*obj.e_r; + closure = obj.Hi*tau*obj.e_r'; + penalty = -obj.Hi*tau*neighbour_scheme.e_l'; + end + + end + + function N = size(obj) + N = obj.m; + end + + end + + methods(Static) + % Calculates the matrices needed for the inteface coupling between boundary bound_u of scheme schm_u + % and bound_v of scheme schm_v. + % [uu, uv, vv, vu] = inteface_coupling(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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+scheme/Staggered1DAcousticsVariable.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,254 @@ +classdef Staggered1DAcousticsVariable < scheme.Scheme + properties + m % Number of points of primal grid in each direction, possibly a vector + h % Grid spacing + + % Grids + grid % Total grid object + grid_primal + grid_dual + + order % Order accuracy for the approximation + + H % Combined norm + Hi % Inverse + D % Semi-discrete approximation matrix + + D1_primal + D1_dual + + % Pick out left or right boundary + e_l + e_r + + % Initial data + v0 + + % Pick out primal or dual component + e_primal + e_dual + + % System matrices, cell matrices of function handles + A + B + + % Variable coefficient matrices evaluated at boundaries. + A_l, A_r + B_l, B_r + end + + + methods + % Scheme for A*u_t + B u_x = 0, + % u = [p, v]; + % A: Diagonal and A > 0, + % A = [a1, 0; + % 0, a2] + % + % B = B^T and with diagonal entries = 0. + % B = [0 b + % b 0] + % Here we store p on the primal grid and v on the dual + % A and B are cell matrices of function handles + function obj = Staggered1DAcousticsVariable(g, order, A, B) + default_arg('B',{@(x)0*x, @(x)0*x+1; @(x)0*x+1, @(x)0*x}); + default_arg('A',{@(x)0*x+1, @(x)0*x; @(x)0*x, @(x)0*x+1}); + + obj.order = order; + obj.A = A; + obj.B = B; + + % Grids + obj.m = g.size(); + xl = g.getBoundary('l'); + xr = g.getBoundary('r'); + xlim = {xl, xr}; + obj.grid = g; + obj.grid_primal = g.grids{1}; + obj.grid_dual = g.grids{2}; + x_primal = obj.grid_primal.points(); + x_dual = obj.grid_dual.points(); + + % Boundary matrices + obj.A_l = [A{1,1}(xl), A{1,2}(xl);.... + A{2,1}(xl), A{2,2}(xl)]; + obj.A_r = [A{1,1}(xr), A{1,2}(xr);.... + A{2,1}(xr), A{2,2}(xr)]; + obj.B_l = [B{1,1}(xl), B{1,2}(xl);.... + B{2,1}(xl), B{2,2}(xl)]; + obj.B_r = [B{1,1}(xr), B{1,2}(xr);.... + B{2,1}(xr), B{2,2}(xr)]; + + % Get operators + ops = sbp.D1StaggeredUpwind(obj.m, xlim, order); + obj.h = ops.h; + + % Build combined operators + H_primal = ops.H_primal; + H_dual = ops.H_dual; + obj.H = blockmatrix.toMatrix( {H_primal, []; [], H_dual } ); + obj.Hi = inv(obj.H); + + D1_primal = ops.D1_primal; + D1_dual = ops.D1_dual; + A11_B12 = spdiag(-1./A{1,1}(x_primal).*B{1,2}(x_primal), 0); + A22_B21 = spdiag(-1./A{2,2}(x_dual).*B{2,1}(x_dual), 0); + D = {[], A11_B12*D1_primal;... + A22_B21*D1_dual, []}; + obj.D = blockmatrix.toMatrix(D); + obj.D1_primal = D1_primal; + obj.D1_dual = D1_dual; + + % Combined boundary operators + e_primal_l = ops.e_primal_l; + e_primal_r = ops.e_primal_r; + e_dual_l = ops.e_dual_l; + e_dual_r = ops.e_dual_r; + e_l = {e_primal_l, [];... + [] , e_dual_l}; + e_r = {e_primal_r, [];... + [] , e_dual_r}; + obj.e_l = blockmatrix.toMatrix(e_l); + obj.e_r = blockmatrix.toMatrix(e_r); + + % Pick out first or second component of solution + N_primal = obj.grid_primal.N(); + N_dual = obj.grid_dual.N(); + obj.e_primal = [speye(N_primal, N_primal); sparse(N_dual, N_primal)]; + obj.e_dual = [sparse(N_primal, N_dual); speye(N_dual, N_dual)]; + + + end + % Closure functions return the operators applied to the own domain to close the boundary + % Penalty functions return the operators to force the solution. In the case of an interface it returns the operator applied to the other domain. + % 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','p'); + + % type = 'p' => boundary condition for p + % type = 'v' => boundary condition for v + % No other types implemented yet + + % BC on the form Lu - g = 0; + + % Need a transformation T such that w = T^{−1}*u is a + % meaningful change of variables and T^T*B*T is block-diagonal, + % For linear acoustics, T = T_C meets these criteria. + + % Get boundary matrices + switch boundary + case 'l' + A = obj.A_l; + B = obj.B_l; + case 'r' + A = obj.A_r; + B = obj.B_r; + end + C = inv(A)*B; + + % Diagonalize C and use T_C to diagonalize B + [T, ~] = eig(C); + Lambda = T'*B*T; + lambda = diag(Lambda); + + % Identify in- and outgoing characteristic variables + Iplus = lambda > 0; + Iminus = lambda < 0; + + switch boundary + case 'l' + Iin = Iplus; + case 'r' + Iin = Iminus; + end + Tin = T(:,Iin); + + switch type + case 'p' + L = [1, 0]; + case 'v' + L = [0, 1]; + case 'characteristic' + % Diagonalize C + [T_C, Lambda_C] = eig(C); + lambda_C = diag(Lambda_C); + % Identify in- and outgoing characteristic variables + Iplus = lambda_C > 0; + Iminus = lambda_C < 0; + + switch boundary + case 'l' + Iin_C = Iplus; + case 'r' + Iin_C = Iminus; + end + T_C_inv = inv(T_C); + L = T_C_inv(Iin_C,:); + otherwise + error('Boundary condition not implemented.'); + end + + % Penalty parameters + sigma = [0; 0]; + sigma(Iin) = lambda(Iin); + + % Sparsify + A = sparse(A); + T = sparse(T); + sigma = sparse(sigma); + L = sparse(L); + Tin = sparse(Tin); + + switch boundary + case 'l' + tau = -1*obj.e_l * inv(A) * inv(T)' * sigma * inv(L*Tin); + closure = obj.Hi*tau*L*obj.e_l'; + + case 'r' + tau = 1*obj.e_r * inv(A) * inv(T)' * sigma * inv(L*Tin); + closure = obj.Hi*tau*L*obj.e_r'; + + end + + penalty = -obj.Hi*tau; + + end + + function [closure, penalty] = interface(obj,boundary,neighbour_scheme,neighbour_boundary) + + error('Staggered1DAcoustics, interface not implemented'); + + switch boundary + % Upwind coupling + case {'l','left'} + tau = -1*obj.e_l; + closure = obj.Hi*tau*obj.e_l'; + penalty = -obj.Hi*tau*neighbour_scheme.e_r'; + case {'r','right'} + tau = 0*obj.e_r; + closure = obj.Hi*tau*obj.e_r'; + penalty = -obj.Hi*tau*neighbour_scheme.e_l'; + end + + end + + function N = size(obj) + N = obj.m; + end + + end + + methods(Static) + % Calculates the matrices needed for the inteface coupling between boundary bound_u of scheme schm_u + % and bound_v of scheme schm_v. + % [uu, uv, vv, vu] = inteface_coupling(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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+time/+rkparameters/rk4.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,12 @@ +function [a,b,c,s] = rk4() + +% Butcher tableau for classical RK$ +s = 4; +a = sparse(s,s); +a(2,1) = 1/2; +a(3,2) = 1/2; +a(4,3) = 1; +b = 1/6*[1; 2; 2; 1]; +c = [0; 1/2; 1/2; 1]; + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+time/ExplicitRungeKuttaDiscreteData.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,116 @@ +classdef ExplicitRungeKuttaDiscreteData < time.Timestepper + properties + D + S % Function handle for time-dependent data + data % Matrix of data vectors, one column per stage + F + k + t + v + m + n + order + a, b, c, s % Butcher tableau + K % Stage rates + U % Stage approximations + T % Stage times + end + + + methods + function obj = ExplicitRungeKuttaDiscreteData(D, S, data, k, t0, v0, order) + default_arg('order', 4); + default_arg('S', []); + default_arg('data', []); + + obj.D = D; + obj.S = S; + obj.k = k; + obj.t = t0; + obj.v = v0; + obj.m = length(v0); + obj.n = 0; + obj.order = order; + obj.data = data; + + switch order + case 4 + [obj.a, obj.b, obj.c, obj.s] = time.rkparameters.rk4(); + otherwise + error('That RK method is not available'); + end + + obj.K = sparse(obj.m, obj.s); + obj.U = sparse(obj.m, obj.s); + + end + + function [v,t,U,T,K] = getV(obj) + v = obj.v; + t = obj.t; + U = obj.U; % Stage approximations in previous time step. + T = obj.T; % Stage times in previous time step. + K = obj.K; % Stage rates in previous time step. + end + + function [a,b,c,s] = getTableau(obj) + a = obj.a; + b = obj.b; + c = obj.c; + s = obj.s; + end + + function obj = step(obj) + v = obj.v; + a = obj.a; + b = obj.b; + c = obj.c; + s = obj.s; + S = obj.S; + dt = obj.k; + K = obj.K; + U = obj.U; + D = obj.D; + data = obj.data; + + for i = 1:s + U(:,i) = v; + for j = 1:i-1 + U(:,i) = U(:,i) + dt*a(i,j)*K(:,j); + end + + K(:,i) = D*U(:,i); + obj.T(i) = obj.t + c(i)*dt; + + % Data from continuos function and discrete time-points. + if ~isempty(S) + K(:,i) = K(:,i) + S(obj.T(i)); + end + if ~isempty(data) + K(:,i) = K(:,i) + data(:,obj.n*s + i); + end + + end + + obj.v = v + dt*K*b; + obj.t = obj.t + dt; + obj.n = obj.n + 1; + obj.U = U; + obj.K = K; + end + end + + + methods (Static) + function k = getTimeStep(lambda) + + switch obj.order + case 4 + k = rk4.get_rk4_time_step(lambda); + otherwise + error('Time-step function not available for this order'); + end + end + end + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diracDiscr.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,97 @@ +function ret = diracDiscr(x_0in , x , m_order, s_order, H, h) + +m = length(x); + +% Return zeros if x0 is outside grid +if(x_0in < x(1) || x_0in > x(end) ) + + ret = zeros(size(x)); + +else + + fnorm = diag(H); + eta = abs(x-x_0in); + tot = m_order+s_order; + S = []; + M = []; + + % Get interior grid spacing + middle = floor(m/2); + h = x(middle+1) - x(middle); + + poss = find(tot*h/2 >= eta); + + % Ensure that poss is not too long + if length(poss) == (tot + 2) + poss = poss(2:end-1); + elseif length(poss) == (tot + 1) + poss = poss(1:end-1); + end + + % Use first tot grid points + if length(poss)<tot && x_0in < x(1) + ceil(tot/2)*h; + index=1:tot; + pol=(x(1:tot)-x(1))/(x(tot)-x(1)); + x_0=(x_0in-x(1))/(x(tot)-x(1)); + norm=fnorm(1:tot)/h; + + % Use last tot grid points + elseif length(poss)<tot && x_0in > x(end) - ceil(tot/2)*h; + index = length(x)-tot+1:length(x); + pol = (x(end-tot+1:end)-x(end-tot+1))/(x(end)-x(end-tot+1)); + norm = fnorm(end-tot+1:end)/h; + x_0 = (x_0in-x(end-tot+1))/(x(end)-x(end-tot+1)); + + % Interior, compensate for round-off errors. + elseif length(poss) < tot + if poss(end)<m + poss = [poss; poss(end)+1]; + else + poss = [poss(1)-1; poss]; + end + pol = (x(poss)-x(poss(1)))/(x(poss(end))-x(poss(1))); + x_0 = (x_0in-x(poss(1)))/(x(poss(end))-x(poss(1))); + norm = fnorm(poss)/h; + index = poss; + + % Interior + else + pol = (x(poss)-x(poss(1)))/(x(poss(end))-x(poss(1))); + x_0 = (x_0in-x(poss(1)))/(x(poss(end))-x(poss(1))); + norm = fnorm(poss)/h; + index = poss; + end + + h_pol = pol(2)-pol(1); + b = zeros(m_order+s_order,1); + + for i = 1:m_order + b(i,1) = x_0^(i-1); + end + + for i = 1:(m_order+s_order) + for j = 1:m_order + M(j,i) = pol(i)^(j-1)*h_pol*norm(i); + end + end + + for i = 1:(m_order+s_order) + for j = 1:s_order + S(j,i) = (-1)^(i-1)*pol(i)^(j-1); + end + end + + A = [M;S]; + + d = A\b; + ret = x*0; + ret(index) = d/h*h_pol; +end + +end + + + + + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/diracDiscrTest.m Fri Jun 15 18:10:26 2018 -0700 @@ -0,0 +1,379 @@ +function tests = diracDiscrTest() + tests = functiontests(localfunctions); +end + +function testLeftGP(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test left boundary grid points + x0s = xl + [0, h, 2*h]; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testLeftRandom(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test random points near left boundary + x0s = xl + 2*h*rand(1,10); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testRightGP(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test right boundary grid points + x0s = xr-[0, h, 2*h]; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testRightRandom(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test random points near right boundary + x0s = xr - 2*h*rand(1,10); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testInteriorGP(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test interior grid points + m_half = round(m/2); + x0s = xl + (m_half-1:m_half+1)*h; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testInteriorRandom(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test random points in interior + x0s = (xl+2*h) + (xr-xl-4*h)*rand(1,20); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +% x0 outside grid should yield 0 integral! +function testX0OutsideGrid(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test points outisde grid + x0s = [xl-1.1*h, xr+1.1*h]; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - 0); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testAllGP(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test all grid points + x0s = x; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testHalfGP(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond); + + % Test halfway between all grid points + x0s = 1/2*( x(2:end)+x(1:end-1) ); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testAllGPStaggered(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStaggered(order, mom_cond); + + % Test all grid points + x0s = x; + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testHalfGPStaggered(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStaggered(order, mom_cond); + + % Test halfway between all grid points + x0s = 1/2*( x(2:end)+x(1:end-1) ); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + +function testRandomStaggered(testCase) + + orders = [2, 4, 6]; + mom_conds = orders; + + for o = 1:length(orders) + order = orders(o); + mom_cond = mom_conds(o); + [xl, xr, m, h, x, H, fs] = setupStaggered(order, mom_cond); + + % Test random points within grid boundaries + x0s = xl + (xr-xl)*rand(1,300); + + for j = 1:length(fs) + f = fs{j}; + fx = f(x); + for i = 1:length(x0s) + x0 = x0s(i); + delta = diracDiscr(x0, x, mom_cond, 0, H); + integral = delta'*H*fx; + err = abs(integral - f(x0)); + testCase.verifyLessThan(err, 1e-12); + end + end + end +end + + +% ============== Setup functions ======================= +function [xl, xr, m, h, x, H, fs] = setupStuff(order, mom_cond) + + % Grid + xl = -3; + xr = 900; + L = xr-xl; + m = 101; + h = (xr-xl)/(m-1); + g = grid.equidistant(m, {xl, xr}); + x = g.points(); + + % Quadrature + ops = sbp.D2Standard(m, {xl, xr}, order); + H = ops.H; + + % Moment conditions + fs = cell(mom_cond,1); + for p = 0:mom_cond-1 + fs{p+1} = @(x) (x/L).^p; + end + +end + +function [xl, xr, m, h, x, H, fs] = setupStaggered(order, mom_cond) + + % Grid + xl = -3; + xr = 900; + L = xr-xl; + m = 101; + [~, g_dual] = grid.primalDual1D(m, {xl, xr}); + x = g_dual.points(); + h = g_dual.h; + + % Quadrature + ops = sbp.D1Staggered(m, {xl, xr}, order); + H = ops.H_dual; + + % Moment conditions + fs = cell(mom_cond,1); + for p = 0:mom_cond-1 + fs{p+1} = @(x) (x/L).^p; + end + +end + +