Mercurial > repos > public > sbplib
changeset 29:32b39dc44474
Removed repository inside +sbp to make it part of the root repo.
| author | Jonatan Werpers <jonatan@werpers.com> |
|---|---|
| date | Mon, 28 Sep 2015 08:47:28 +0200 |
| parents | 16acb2775aca |
| children | cd2e28c5ecd2 |
| files | +sbp/BlockNorm.m +sbp/Higher.m +sbp/HigherCompatible.m +sbp/HigherCompatibleVariable.m +sbp/HigherPeriodic.m +sbp/OpSet.m +sbp/Ordinary.m +sbp/Variable.m +sbp/blocknorm10.m +sbp/blocknorm4.m +sbp/blocknorm6.m +sbp/blocknorm8.m +sbp/diagInd.m +sbp/higher2_compatible_halfvariable.m +sbp/higher4.m +sbp/higher4_compatible_halfvariable.m +sbp/higher6.m +sbp/higher6_compatible_halfvariable.m +sbp/higher_compatible2.m +sbp/higher_compatible4.m +sbp/higher_compatible6.m +sbp/ordinary10.m +sbp/ordinary2.m +sbp/ordinary4.m +sbp/ordinary6.m +sbp/ordinary8.m +sbp/variable4.m |
| diffstat | 27 files changed, 2311 insertions(+), 0 deletions(-) [+] |
line wrap: on
line diff
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/BlockNorm.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,55 @@ +classdef BlockNorm < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = BlockNorm(m,h,order) + + if order == 4 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.blocknorm4(m,h); + elseif order == 6 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.blocknorm6(m,h); + elseif order == 8 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.blocknorm8(m,h); + elseif order == 10 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.blocknorm10(m,h); + else + error('Invalid operator order %d.',order); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.Q = Q; + obj.norms.M = M; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D2 = D2; + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/Higher.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,64 @@ +classdef Higher < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = Higher(m,h,order) + + if order == 4 + [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); + obj.borrowing.N.S2 = 0.5485; + obj.borrowing.N.S3 = 1.0882; + elseif order == 6 + [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); + obj.borrowing.N.S2 = 0.3227; + obj.borrowing.N.S3 = 0.1568; + else + error('Invalid operator order %d.',order); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.Q = Q; + obj.norms.M = M; + obj.norms.Q3 = Q3; + obj.norms.N = M4; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + obj.boundary.S2_1 = S2_1; + obj.boundary.S3_1 = S3_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + obj.boundary.S2_m = S2_m; + obj.boundary.S3_m = S3_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D2 = D2; + obj.derivatives.D3 = D3; + obj.derivatives.D4 = D4; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/HigherCompatible.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,64 @@ +classdef HigherCompatible < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = HigherCompatible(m,h,order) + + if order == 2 + [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); + obj.borrowing.N.S2 = 0.7500; + obj.borrowing.N.S3 = 0.3000; + elseif order == 4 + [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); + obj.borrowing.N.S2 = 0.4210; + obj.borrowing.N.S3 = 0.7080; + elseif order == 6 + [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); + obj.borrowing.N.S2 = 0.06925; + obj.borrowing.N.S3 = 0.05128; + else + error('Invalid operator order.'); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.Q = Q; + obj.norms.N = M4; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + obj.boundary.S2_1 = S2_1; + obj.boundary.S3_1 = S3_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + obj.boundary.S2_m = S2_m; + obj.boundary.S3_m = S3_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D4 = D4; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/HigherCompatibleVariable.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,63 @@ +classdef HigherCompatibleVariable < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = HigherCompatibleVariable(m,h,order) + + if order == 2 + [H, HI, D1, D2, D3, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher2_compatible_halfvariable(m,h); + obj.borrowing.N.S2 = 1.2500; + obj.borrowing.N.S3 = 0.4000; + elseif order == 4 + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher4_compatible_halfvariable(m,h); + obj.borrowing.N.S2 = 0.5055; + obj.borrowing.N.S3 = 0.9290; + elseif order == 6 + [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher6_compatible_halfvariable(m,h); + obj.borrowing.N.S2 = 0.3259; + obj.borrowing.N.S3 = 0.1580; + else + error('Invalid operator order.'); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.N = M4; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + obj.boundary.S2_1 = S2_1; + obj.boundary.S3_1 = S3_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + obj.boundary.S2_m = S2_m; + obj.boundary.S3_m = S3_m; + + obj.derivatives.D2 = D2; + obj.derivatives.D4 = D4; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/HigherPeriodic.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,65 @@ +classdef HigherPeriodic < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + + + methods + function obj = HigherCompatible(m,h,order) + + if order == 2 + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher2_compatible(m,h); + obj.borrowing.N.S2 = 0.7500; + obj.borrowing.N.S3 = 0.3000; + elseif order == 4 + + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher4_compatible(m,h); + obj.borrowing.N.S2 = 0.4210; + obj.borrowing.N.S3 = 0.7080; + elseif order == 6 + [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = sbp.higher6_compatible(m,h); + obj.borrowing.N.S2 = 0.06925; + obj.borrowing.N.S3 = 0.05128; + else + error('Invalid operator order.'); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.Q = Q; + obj.norms.N = M4; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + obj.boundary.S2_1 = S2_1; + obj.boundary.S3_1 = S3_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + obj.boundary.S2_m = S2_m; + obj.boundary.S3_m = S3_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D4 = D4; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end + + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/OpSet.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,18 @@ +classdef (Abstract) OpSet + properties (Abstract) + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + methods (Abstract) + + end + + methods (Abstract, Static) + lambda = smallestGrid() + end +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/Ordinary.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,63 @@ +classdef Ordinary < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + methods + function obj = Ordinary(m,h,order) + + if order == 2 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.ordinary2(m,h); + obj.borrowing.M.S = 0.4000; + elseif order == 4 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.ordinary4(m,h); + obj.borrowing.M.S = 0.2508; + elseif order == 6 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.ordinary6(m,h); + obj.borrowing.M.S = 0.1878; + elseif order == 8 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.ordinary8(m,h); + obj.borrowing.M.S = 0.0015; + elseif order == 10 + [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = sbp.ordinary10(m,h); + obj.borrowing.M.S = 0.0351; + else + error('Invalid operator order %d.',order); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + obj.norms.Q = Q; + obj.norms.M = M; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D2 = D2; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end +end + + + + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/Variable.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,52 @@ +classdef Variable < sbp.OpSet + properties + norms % Struct containing norm matrices such as H,Q, M + boundary % Struct contanging vectors for boundry point approximations + derivatives % Struct containging differentiation operators + borrowing % Struct with borrowing limits for different norm matrices + m % Number of grid points. + h % Step size + end + + methods + function obj = Variable(m,h,order) + + switch order + case 4 + [H, HI, D1, D2, e_1, e_m, S_1, S_m] = sbp.variable4(m,h); + obj.borrowing.M.S = 0.2505765857; + otherwise + error('Invalid operator order %d.',order); + end + + obj.h = h; + obj.m = m; + + obj.norms.H = H; + obj.norms.HI = HI; + % obj.norms.Q = Q; + % obj.norms.M = M; + + obj.boundary.e_1 = e_1; + obj.boundary.S_1 = S_1; + + obj.boundary.e_m = e_m; + obj.boundary.S_m = S_m; + + obj.derivatives.D1 = D1; + obj.derivatives.D2 = D2; + + end + end + + methods (Static) + function lambda = smallestGrid(obj) + error('Not implmented') + end + end +end + + + + +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/blocknorm10.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,57 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = blocknorm10(m,h) + H_U=[0.428081020217e12 / 0.2633637888000e13 0.779032713983e12 / 0.2633637888000e13 -0.1187642619571e13 / 0.2633637888000e13 0.1642279196603e13 / 0.2633637888000e13 -0.339289243121e12 / 0.526727577600e12 0.1261055176253e13 / 0.2633637888000e13 -0.658216413073e12 / 0.2633637888000e13 0.33968779823e11 / 0.376233984000e12 -0.764998e6 / 0.40186125e8; 0.779032713983e12 / 0.2633637888000e13 0.317907052061e12 / 0.164602368000e12 -0.1082918052397e13 / 0.658409472000e12 0.176473501369e12 / 0.82301184000e11 -0.521625191587e12 / 0.263363788800e12 0.200523313337e12 / 0.164602368000e12 -0.279496000009e12 / 0.658409472000e12 -0.832e3 / 0.40186125e8 0.129794760887e12 / 0.2633637888000e13; -0.1187642619571e13 / 0.2633637888000e13 -0.1082918052397e13 / 0.658409472000e12 0.1304108863849e13 / 0.329204736000e12 -0.533093695961e12 / 0.131681894400e12 0.213949854133e12 / 0.52672757760e11 -0.1883976009151e13 / 0.658409472000e12 0.51084128e8 / 0.40186125e8 -0.92763684343e11 / 0.658409472000e12 -0.59058717923e11 / 0.526727577600e12; 0.1642279196603e13 / 0.2633637888000e13 0.176473501369e12 / 0.82301184000e11 -0.533093695961e12 / 0.131681894400e12 0.217677310051e12 / 0.32920473600e11 -0.1509120465127e13 / 0.263363788800e12 0.170839232e9 / 0.40186125e8 -0.1404096707137e13 / 0.658409472000e12 0.11789520859e11 / 0.32920473600e11 0.85652315431e11 / 0.526727577600e12; -0.339289243121e12 / 0.526727577600e12 -0.521625191587e12 / 0.263363788800e12 0.213949854133e12 / 0.52672757760e11 -0.1509120465127e13 / 0.263363788800e12 0.21849109e8 / 0.3214890e7 -0.1134422468377e13 / 0.263363788800e12 0.602448430967e12 / 0.263363788800e12 -0.4910542309e10 / 0.10534551552e11 -0.83039945231e11 / 0.526727577600e12; 0.1261055176253e13 / 0.2633637888000e13 0.200523313337e12 / 0.164602368000e12 -0.1883976009151e13 / 0.658409472000e12 0.170839232e9 / 0.40186125e8 -0.1134422468377e13 / 0.263363788800e12 0.681437038097e12 / 0.164602368000e12 -0.1108257453763e13 / 0.658409472000e12 0.31631872327e11 / 0.82301184000e11 0.37820115539e11 / 0.376233984000e12; -0.658216413073e12 / 0.2633637888000e13 -0.279496000009e12 / 0.658409472000e12 0.51084128e8 / 0.40186125e8 -0.1404096707137e13 / 0.658409472000e12 0.602448430967e12 / 0.263363788800e12 -0.1108257453763e13 / 0.658409472000e12 0.623491124887e12 / 0.329204736000e12 -0.146643738067e12 / 0.658409472000e12 -0.98874149197e11 / 0.2633637888000e13; 0.33968779823e11 / 0.376233984000e12 -0.832e3 / 0.40186125e8 -0.92763684343e11 / 0.658409472000e12 0.11789520859e11 / 0.32920473600e11 -0.4910542309e10 / 0.10534551552e11 0.31631872327e11 / 0.82301184000e11 -0.146643738067e12 / 0.658409472000e12 0.174599973347e12 / 0.164602368000e12 0.4625165773e10 / 0.526727577600e12; -0.764998e6 / 0.40186125e8 0.129794760887e12 / 0.2633637888000e13 -0.59058717923e11 / 0.526727577600e12 0.85652315431e11 / 0.526727577600e12 -0.83039945231e11 / 0.526727577600e12 0.37820115539e11 / 0.376233984000e12 -0.98874149197e11 / 0.2633637888000e13 0.4625165773e10 / 0.526727577600e12 0.525286231387e12 / 0.526727577600e12;]; + + + H=eye(m); + H(1:9,1:9)=H_U; + H(m-8:m,m-8:m)=flipud( fliplr(H_U(1:9,1:9) ) ); + H=H*h; + HI=inv(H); + + Q=1/1260*diag(ones(m-5,1),5)-5/504*diag(ones(m-4,1),4)+5/84*diag(ones(m-3,1),3)-5/21*diag(ones(m-2,1),2)+5/6*diag(ones(m-1,1),1)-5/6*diag(ones(m-1,1),-1)+5/21*diag(ones(m-2,1),-2)-5/84*diag(ones(m-3,1),-3)+5/504*diag(ones(m-4,1),-4)-1/1260*diag(ones(m-5,1),-5) ; + + Q_U = [-0.1e1 / 0.2e1 0.78249683e8 / 0.71442000e8 -0.28290472447e11 / 0.18289152000e11 0.4285528063e10 / 0.2032128000e10 -0.3924872557e10 / 0.1828915200e10 0.2856344621e10 / 0.1828915200e10 -0.1598284927e10 / 0.2032128000e10 0.4652402687e10 / 0.18289152000e11 -0.731623e6 / 0.17860500e8; -0.78249683e8 / 0.71442000e8 0 0.5351945471e10 / 0.2032128000e10 -0.4078428731e10 / 0.1143072000e10 0.4879509877e10 / 0.1219276800e10 -0.1273709579e10 / 0.381024000e9 0.7433128649e10 / 0.3657830400e10 -0.1239008e7 / 0.1488375e7 0.21251483e8 / 0.124416000e9; 0.28290472447e11 / 0.18289152000e11 -0.5351945471e10 / 0.2032128000e10 0 0.6701592799e10 / 0.2612736000e10 -0.57535927e8 / 0.17418240e8 0.3058732543e10 / 0.870912000e9 -0.28821953e8 / 0.10206000e8 0.1858901437e10 / 0.1219276800e10 -0.2363118211e10 / 0.6096384000e10; -0.4285528063e10 / 0.2032128000e10 0.4078428731e10 / 0.1143072000e10 -0.6701592799e10 / 0.2612736000e10 0 0.52016695e8 / 0.20901888e8 -0.3777923e7 / 0.1275750e7 0.123333949e9 / 0.41472000e8 -0.2256321727e10 / 0.1143072000e10 0.1058261459e10 / 0.1828915200e10; 0.3924872557e10 / 0.1828915200e10 -0.4879509877e10 / 0.1219276800e10 0.57535927e8 / 0.17418240e8 -0.52016695e8 / 0.20901888e8 0 0.125390297e9 / 0.58060800e8 -0.609569351e9 / 0.261273600e9 0.438488399e9 / 0.243855360e9 -0.14244569e8 / 0.24385536e8; -0.2856344621e10 / 0.1828915200e10 0.1273709579e10 / 0.381024000e9 -0.3058732543e10 / 0.870912000e9 0.3777923e7 / 0.1275750e7 -0.125390297e9 / 0.58060800e8 0 0.4624381729e10 / 0.2612736000e10 -0.484096919e9 / 0.381024000e9 0.2676438019e10 / 0.6096384000e10; 0.1598284927e10 / 0.2032128000e10 -0.7433128649e10 / 0.3657830400e10 0.28821953e8 / 0.10206000e8 -0.123333949e9 / 0.41472000e8 0.609569351e9 / 0.261273600e9 -0.4624381729e10 / 0.2612736000e10 0 0.21500967689e11 / 0.18289152000e11 -0.7199454721e10 / 0.18289152000e11; -0.4652402687e10 / 0.18289152000e11 0.1239008e7 / 0.1488375e7 -0.1858901437e10 / 0.1219276800e10 0.2256321727e10 / 0.1143072000e10 -0.438488399e9 / 0.243855360e9 0.484096919e9 / 0.381024000e9 -0.21500967689e11 / 0.18289152000e11 0 0.761653e6 / 0.882000e6; 0.731623e6 / 0.17860500e8 -0.21251483e8 / 0.124416000e9 0.2363118211e10 / 0.6096384000e10 -0.1058261459e10 / 0.1828915200e10 0.14244569e8 / 0.24385536e8 -0.2676438019e10 / 0.6096384000e10 0.7199454721e10 / 0.18289152000e11 -0.761653e6 / 0.882000e6 0;]; + + Q(1:9,1:9)=Q_U; + Q(m-8:m,m-8:m)=flipud( fliplr(-Q_U(1:9,1:9) ) ); + + D1=HI*Q; + + M_U =[0.3812926003e10 / 0.2438553600e10 -0.5433856529e10 / 0.1741824000e10 0.4187462879e10 / 0.1045094400e10 -0.65635105447e11 / 0.12192768000e11 0.1457682577e10 / 0.270950400e9 -0.27884016067e11 / 0.7315660800e10 0.22304839493e11 / 0.12192768000e11 -0.188132543e9 / 0.348364800e9 0.42711619e8 / 0.571536000e9; -0.5433885329e10 / 0.1741824000e10 0.23985229969e11 / 0.2286144000e10 -0.10208460799e11 / 0.571536000e9 0.8828370001e10 / 0.381024000e9 -0.12306735263e11 / 0.522547200e9 0.39313626089e11 / 0.2286144000e10 -0.4380200287e10 / 0.508032000e9 0.192498023e9 / 0.71442000e8 -0.14565232681e11 / 0.36578304000e11; 0.29313778073e11 / 0.7315660800e10 -0.10209211399e11 / 0.571536000e9 0.24157533391e11 / 0.653184000e9 -0.6561725111e10 / 0.130636800e9 0.26490755639e11 / 0.522547200e9 -0.352801289e9 / 0.9331200e7 0.401374423e9 / 0.20412000e8 -0.29541854057e11 / 0.4572288000e10 0.7396364989e10 / 0.7315660800e10; -0.65653163047e11 / 0.12192768000e11 0.8832999601e10 / 0.381024000e9 -0.6566554871e10 / 0.130636800e9 0.1575758731e10 / 0.21772800e8 -0.2571648133e10 / 0.34836480e8 0.556969019e9 / 0.10206000e8 -0.3139265911e10 / 0.108864000e9 0.165424529e9 / 0.16934400e8 -0.11623567549e11 / 0.7315660800e10; 0.1458632977e10 / 0.270950400e9 -0.86260417241e11 / 0.3657830400e10 0.3792749777e10 / 0.74649600e8 -0.2576699653e10 / 0.34836480e8 0.157840723e9 / 0.2041200e7 -0.29838889141e11 / 0.522547200e9 0.5142742211e10 / 0.174182400e9 -0.36792522023e11 / 0.3657830400e10 0.2438136689e10 / 0.1463132160e10; -0.27909676867e11 / 0.7315660800e10 0.39389053289e11 / 0.2286144000e10 -0.2478149663e10 / 0.65318400e8 0.559214069e9 / 0.10206000e8 -0.29914661941e11 / 0.522547200e9 0.2843819551e10 / 0.65318400e8 -0.14816015149e11 / 0.653184000e9 0.1677001673e10 / 0.228614400e9 -0.44441740171e11 / 0.36578304000e11; 0.3188985299e10 / 0.1741824000e10 -0.626864041e9 / 0.72576000e8 0.57539389e8 / 0.2916000e7 -0.3153500311e10 / 0.108864000e9 0.5162239811e10 / 0.174182400e9 -0.14840163949e11 / 0.653184000e9 0.419025709e9 / 0.31104000e8 -0.138945749e9 / 0.27216000e8 0.591880819e9 / 0.746496000e9; -0.1317440441e10 / 0.2438553600e10 0.192696473e9 / 0.71442000e8 -0.4230355151e10 / 0.653184000e9 0.165983249e9 / 0.16934400e8 -0.36905792423e11 / 0.3657830400e10 0.1679779433e10 / 0.228614400e9 -0.972870443e9 / 0.190512000e9 0.9129544111e10 / 0.2286144000e10 -0.13387742111e11 / 0.7315660800e10; 0.42721069e8 / 0.571536000e9 -0.14572922281e11 / 0.36578304000e11 0.7407199549e10 / 0.7315660800e10 -0.11649228349e11 / 0.7315660800e10 0.349038407e9 / 0.209018880e9 -0.44495912971e11 / 0.36578304000e11 0.29009849731e11 / 0.36578304000e11 -0.13387863071e11 / 0.7315660800e10 0.21585797479e11 / 0.7315660800e10;]; + + + T=[-0.1e1 / 0.3150e4 0.5e1 / 0.1008e4 -0.5e1 / 0.126e3 0.5e1 / 0.21e2 -0.5e1 / 0.3e1 0.5269e4 / 0.1800e4 -0.5e1 / 0.3e1 0.5e1 / 0.21e2 -0.5e1 / 0.126e3 0.5e1 / 0.1008e4 -0.1e1 / 0.3150e4;]; + M=(T(1)*diag(ones(m-5,1),5)+T(2)*diag(ones(m-4,1),4)+T(3)*diag(ones(m-3,1),3)+T(4)*diag(ones(m-2,1),2)+T(5)*diag(ones(m-1,1),1)+T(7)*diag(ones(m-1,1),-1)+T(8)*diag(ones(m-2,1),-2)+T(9)*diag(ones(m-3,1),-3)+T(10)*diag(ones(m-4,1),-4)+T(11)*diag(ones(m-5,1),-5)+T(6)*diag(ones(m,1),0)); + + M(1:9,1:9)=M_U; + + M(m-8:m,m-8:m)=flipud( fliplr( M_U ) ); + M=M/h; + + DS_U=[0.761e3 / 0.280e3 -8 14 -0.56e2 / 0.3e1 0.35e2 / 0.2e1 -0.56e2 / 0.5e1 0.14e2 / 0.3e1 -0.8e1 / 0.7e1 0.1e1 / 0.8e1;]; + DS=zeros(m,m); + DS(1,1:9)=DS_U; + DS(m,m-8:m)=fliplr(DS_U); + DS=DS/h; + + D2=HI*(-M+DS); + + % Try adding AD to boundary D1=HI*(Q-D9'*D9) + DD_9=zeros(m); + d9=[-1 9 -36 84 -126 126 -84 36 -9 1];t9=sum(abs(d9));%d9=d9/t9; + DD_9(1:1,1:10)=[d9]; + DD_9(m:m,m-9:m)=[d9]; + + + ADD=30*h/(t9)*DD_9'*DD_9; + + e_1 = zeros(m,1); + e_1(1)= 1; + e_m = zeros(m,1); + e_m(end)= 1; + S_1 = -DS(1,:)'; + S_m = DS(end,:)'; + + Q = H*D1-(-e_1*e_1' + e_m*e_m'); + M = -(H*D2-(-e_1*S_1' + e_m*S_m')); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/blocknorm4.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,57 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = blocknorm4(m,h) + H_U=[0.751e3 / 0.3456e4 0.661e3 / 0.3456e4 -0.515e3 / 0.3456e4 0.5e1 / 0.128e3; 0.661e3 / 0.3456e4 0.1405e4 / 0.1152e4 -0.3e1 / 0.128e3 0.29e2 / 0.3456e4; -0.515e3 / 0.3456e4 -0.3e1 / 0.128e3 0.989e3 / 0.1152e4 0.149e3 / 0.3456e4; 0.5e1 / 0.128e3 0.29e2 / 0.3456e4 0.149e3 / 0.3456e4 0.3407e4 / 0.3456e4;]; + + + H=eye(m); + H(1:4,1:4)=H_U; + H(m-3:m,m-3:m)=flipud( fliplr(H_U(1:4,1:4) ) ); + H=H*h; + HI=inv(H); + + 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 = [-0.1e1 / 0.2e1 0.55e2 / 0.72e2 -0.47e2 / 0.144e3 0.1e1 / 0.16e2; -0.55e2 / 0.72e2 0 0.43e2 / 0.48e2 -0.19e2 / 0.144e3; 0.47e2 / 0.144e3 -0.43e2 / 0.48e2 0 0.47e2 / 0.72e2; -0.1e1 / 0.16e2 0.19e2 / 0.144e3 -0.47e2 / 0.72e2 0;]; + + Q(1:4,1:4)=Q_U; + Q(m-3:m,m-3:m)=flipud( fliplr(-Q_U(1:4,1:4) ) ); + + D1=HI*Q; + + M_U=[0.359e3 / 0.288e3 -0.443e3 / 0.288e3 0.97e2 / 0.288e3 -0.13e2 / 0.288e3; -0.51e2 / 0.32e2 0.325e3 / 0.96e2 -0.191e3 / 0.96e2 0.19e2 / 0.96e2; 0.43e2 / 0.96e2 -0.69e2 / 0.32e2 0.293e3 / 0.96e2 -0.137e3 / 0.96e2; -0.29e2 / 0.288e3 0.89e2 / 0.288e3 -0.427e3 / 0.288e3 0.727e3 / 0.288e3;]; + + + + 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(1:4,1:4)=M_U; + + M(m-3:m,m-3:m)=flipud( fliplr( M_U ) ); + M=M/h; + + DS_U=[0.25e2 / 0.12e2 -4 3 -0.4e1 / 0.3e1 0.1e1 / 0.4e1;]; + DS=zeros(m,m); + DS(1,1:5)=DS_U; + DS(m,m-4:m)=fliplr(DS_U); + DS=DS/h; + + D2=HI*(-M+DS); + + d3=[-1 3 -3 1]; + t3=sum(abs(d3)); + DD_3(1:1,1:4)=[d3]; + DD_3(m:m,m-3:m)=[d3]; + + % This works for wave eq. + % For studs interface in 1D no AD is needed. + ADD=1*h/(t3)*DD_3'*DD_3; + + e_1 = zeros(m,1); + e_1(1)= 1; + e_m = zeros(m,1); + e_m(end)= 1; + S_1 = -DS(1,:)'; + S_m = DS(end,:)'; + + Q = H*D1-(-e_1*e_1' + e_m*e_m'); + M = -(H*D2-(-e_1*S_1' + e_m*S_m')); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/blocknorm6.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,57 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = blocknorm6(m,h) + H_U=[0.8489084265971e13 / 0.45952647390720e14 0.24636450459943e14 / 0.98469958694400e14 -0.2796787072531e13 / 0.12308744836800e14 0.2793599068823e13 / 0.14360202309600e14 -0.66344569931569e14 / 0.689289710860800e15 0.3784697867191e13 / 0.137857942172160e15; 0.24636450459943e14 / 0.98469958694400e14 0.27815394775103e14 / 0.19693991738880e14 -0.445601472229e12 / 0.861612138576e12 0.3896159037731e13 / 0.17232242771520e14 -0.866505556741e12 / 0.27571588434432e14 -0.25625418493681e14 / 0.689289710860800e15; -0.2796787072531e13 / 0.12308744836800e14 -0.445601472229e12 / 0.861612138576e12 0.31409405327129e14 / 0.17232242771520e14 -0.1595539040819e13 / 0.3446448554304e13 0.2651608170899e13 / 0.17232242771520e14 0.1434714163381e13 / 0.43080606928800e14; 0.2793599068823e13 / 0.14360202309600e14 0.3896159037731e13 / 0.17232242771520e14 -0.1595539040819e13 / 0.3446448554304e13 0.6984350202787e13 / 0.5744080923840e13 -0.62662743973e11 / 0.861612138576e12 -0.435331581619e12 / 0.12308744836800e14; -0.66344569931569e14 / 0.689289710860800e15 -0.866505556741e12 / 0.27571588434432e14 0.2651608170899e13 / 0.17232242771520e14 -0.62662743973e11 / 0.861612138576e12 0.20320736807807e14 / 0.19693991738880e14 0.1368363924007e13 / 0.98469958694400e14; 0.3784697867191e13 / 0.137857942172160e15 -0.25625418493681e14 / 0.689289710860800e15 0.1434714163381e13 / 0.43080606928800e14 -0.435331581619e12 / 0.12308744836800e14 0.1368363924007e13 / 0.98469958694400e14 0.27414523542149e14 / 0.27571588434432e14;]; + + + H=eye(m); + H(1:6,1:6)=H_U; + H(m-5:m,m-5:m)=flipud( fliplr(H_U(1:6,1:6) ) ); + H=H*h; + HI=inv(H); + + Q=(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)); + + Q_U = [-0.1e1 / 0.2e1 0.151864337282617e15 / 0.172322427715200e15 -0.251539972254817e15 / 0.344644855430400e15 0.61230525943549e14 / 0.114881618476800e15 -0.80987306509439e14 / 0.344644855430400e15 0.697178163343e12 / 0.13785794217216e14; -0.151864337282617e15 / 0.172322427715200e15 0 0.12350422095979e14 / 0.7658774565120e13 -0.78802251164141e14 / 0.68928971086080e14 0.4229407848431e13 / 0.7658774565120e13 -0.5372490790279e13 / 0.38293872825600e14; 0.251539972254817e15 / 0.344644855430400e15 -0.12350422095979e14 / 0.7658774565120e13 0 0.2217674201683e13 / 0.1723224277152e13 -0.13219134462287e14 / 0.22976323695360e14 0.19660399553981e14 / 0.114881618476800e15; -0.61230525943549e14 / 0.114881618476800e15 0.78802251164141e14 / 0.68928971086080e14 -0.2217674201683e13 / 0.1723224277152e13 0 0.62307836637379e14 / 0.68928971086080e14 -0.84068101764193e14 / 0.344644855430400e15; 0.80987306509439e14 / 0.344644855430400e15 -0.4229407848431e13 / 0.7658774565120e13 0.13219134462287e14 / 0.22976323695360e14 -0.62307836637379e14 / 0.68928971086080e14 0 0.44756810052211e14 / 0.57440809238400e14; -0.697178163343e12 / 0.13785794217216e14 0.5372490790279e13 / 0.38293872825600e14 -0.19660399553981e14 / 0.114881618476800e15 0.84068101764193e14 / 0.344644855430400e15 -0.44756810052211e14 / 0.57440809238400e14 0;]; + + Q(1:6,1:6)=Q_U; + Q(m-5:m,m-5:m)=flipud( fliplr(-Q_U(1:6,1:6) ) ); + + D1=HI*Q; + + M_U=[0.960901171090739e15 / 0.689289710860800e15 -0.502032138770899e15 / 0.229763236953600e15 0.493085196645929e15 / 0.344644855430400e15 -0.329491854944251e15 / 0.344644855430400e15 0.89541920186441e14 / 0.229763236953600e15 -0.50617198740721e14 / 0.689289710860800e15; -0.100483015499831e15 / 0.45952647390720e14 0.807564929223191e15 / 0.137857942172160e15 -0.415779274818991e15 / 0.68928971086080e14 0.80693719872887e14 / 0.22976323695360e14 -0.196663473955997e15 / 0.137857942172160e15 0.37943821632959e14 / 0.137857942172160e15; 0.99938177941669e14 / 0.68928971086080e14 -0.84419552767043e14 / 0.13785794217216e14 0.106922123424097e15 / 0.11488161847680e14 -0.223356054245897e15 / 0.34464485543040e14 0.157526160982357e15 / 0.68928971086080e14 -0.10062402380533e14 / 0.22976323695360e14; -0.68310884976863e14 / 0.68928971086080e14 0.17038649985979e14 / 0.4595264739072e13 -0.231397767539273e15 / 0.34464485543040e14 0.232669188399619e15 / 0.34464485543040e14 -0.1657930371065e13 / 0.510584971008e12 0.34774771016773e14 / 0.68928971086080e14; 0.18789143112277e14 / 0.45952647390720e14 -0.213895716727517e15 / 0.137857942172160e15 0.171024751153381e15 / 0.68928971086080e14 -0.8523669967037e13 / 0.2552924855040e13 0.485768751245399e15 / 0.137857942172160e15 -0.229158724354277e15 / 0.137857942172160e15; -0.51766014925489e14 / 0.689289710860800e15 0.202930494289627e15 / 0.689289710860800e15 -0.54332868549353e14 / 0.114881618476800e15 0.180479548146281e15 / 0.344644855430400e15 -0.1146942437956153e16 / 0.689289710860800e15 0.211001773091419e15 / 0.76587745651200e14;]; + + + + M=-(2*diag(ones(m-3,1),3)-27*diag(ones(m-2,1),2)+270*diag(ones(m-1,1),1)+270*diag(ones(m-1,1),-1)-27*diag(ones(m-2,1),-2)+2*diag(ones(m-3,1),-3)-490*diag(ones(m,1),0))/180; + + M(1:6,1:6)=M_U; + + M(m-5:m,m-5:m)=flipud( fliplr( M_U ) ); + M=M/h; + + DS_U=[0.137e3 / 0.60e2 -5 5 -0.10e2 / 0.3e1 0.5e1 / 0.4e1 -0.1e1 / 0.5e1;]; + DS=zeros(m,m); + DS(1,1:6)=DS_U; + DS(m,m-5:m)=fliplr(DS_U); + DS=DS/h; + + D2=HI*(-M+DS); + + d5=[-1 5 -10 10 -5 1]; + t5=sum(abs(d5)); + DD_5(1:1,1:6)=[d5]; + DD_5(m:m,m-5:m)=[d5]; + + % This works for wave eq. + % For studs interface in 1D no AD is needed. + ADD=7*h/(t5)*DD_5'*DD_5; + + e_1 = zeros(m,1); + e_1(1)= 1; + e_m = zeros(m,1); + e_m(end)= 1; + S_1 = -DS(1,:)'; + S_m = DS(end,:)'; + + Q = H*D1-(-e_1*e_1' + e_m*e_m'); + M = -(H*D2-(-e_1*S_1' + e_m*S_m')); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/blocknorm8.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,60 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = blocknorm8(m,h) + % Eigth order + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + H_U=[ 0.704266523e9 / 0.4180377600e10 0.4579586639e10 / 0.16257024000e11 -0.3623870581e10 / 0.9754214400e10 0.12753127559e11 / 0.29262643200e11 -0.3687413731e10 / 0.9754214400e10 0.2169892891e10 / 0.9754214400e10 -0.13224544841e11 / 0.146313216000e12 0.4142047e7 / 0.199065600e9; + 0.4579586639e10 / 0.16257024000e11 0.36543258551e11 / 0.20901888000e11 -0.8235820121e10 / 0.6967296000e10 0.1800520829e10 / 0.1393459200e10 -0.3725834681e10 / 0.4180377600e10 0.2588501879e10 / 0.6967296000e10 -0.10477621e8 / 0.995328000e9 -0.6589395529e10 / 0.146313216000e12; + -0.3623870581e10 / 0.9754214400e10 -0.8235820121e10 / 0.6967296000e10 0.765685439e9 / 0.258048000e9 -0.3254203513e10 / 0.1393459200e10 0.2477267447e10 / 0.1393459200e10 -0.456533e6 / 0.491520e6 0.873043831e9 / 0.6967296000e10 0.4279558279e10 / 0.48771072000e11; + 0.12753127559e11 / 0.29262643200e11 0.1800520829e10 / 0.1393459200e10 -0.3254203513e10 / 0.1393459200e10 0.16428690611e11 / 0.4180377600e10 -0.460605929e9 / 0.199065600e9 0.1897042423e10 / 0.1393459200e10 -0.1151273401e10 / 0.4180377600e10 -0.65906413e8 / 0.650280960e9; + -0.3687413731e10 / 0.9754214400e10 -0.3725834681e10 / 0.4180377600e10 0.2477267447e10 / 0.1393459200e10 -0.460605929e9 / 0.199065600e9 0.11663916211e11 / 0.4180377600e10 -0.307273957e9 / 0.278691840e9 0.117995903e9 / 0.464486400e9 0.95035807e8 / 0.1170505728e10; + 0.2169892891e10 / 0.9754214400e10 0.2588501879e10 / 0.6967296000e10 -0.456533e6 / 0.491520e6 0.1897042423e10 / 0.1393459200e10 -0.307273957e9 / 0.278691840e9 0.17436823e8 / 0.10321920e8 -0.1274455129e10 / 0.6967296000e10 -0.338917493e9 / 0.9754214400e10; + -0.13224544841e11 / 0.146313216000e12 -0.10477621e8 / 0.995328000e9 0.873043831e9 / 0.6967296000e10 -0.1151273401e10 / 0.4180377600e10 0.117995903e9 / 0.464486400e9 -0.1274455129e10 / 0.6967296000e10 0.22041718711e11 / 0.20901888000e11 0.468461293e9 / 0.48771072000e11; + 0.4142047e7 / 0.199065600e9 -0.6589395529e10 / 0.146313216000e12 0.4279558279e10 / 0.48771072000e11 -0.65906413e8 / 0.650280960e9 0.95035807e8 / 0.1170505728e10 -0.338917493e9 / 0.9754214400e10 0.468461293e9 / 0.48771072000e11 0.20832744839e11 / 0.20901888000e11; + ]; + + + H=eye(m); + H(1:8,1:8)=H_U; + H(m-7:m,m-7:m)=rot90( H_U(1:8,1:8) ,2 ); + H=H*h; + HI=inv(H); + + Q=-(1/280*diag(ones(m-4,1),4)-4/105*diag(ones(m-3,1),3)+1/5*diag(ones(m-2,1),2)-4/5*diag(ones(m-1,1),1)+4/5*diag(ones(m-1,1),-1)-1/5*diag(ones(m-2,1),-2)+4/105*diag(ones(m-3,1),-3)-1/280*diag(ones(m-4,1),-4)); + + + Q_U = [-0.1e1 / 0.2e1 0.16262381e8 / 0.15876000e8 -0.3770744693e10 / 0.3048192000e10 0.290431859e9 / 0.203212800e9 -0.363704879e9 / 0.304819200e9 0.206906927e9 / 0.304819200e9 -0.248883679e9 / 0.1016064000e10 0.2665637e7 / 0.62208000e8; -0.16262381e8 / 0.15876000e8 0 0.138589901e9 / 0.62208000e8 -0.31764881e8 / 0.12441600e8 0.461249e6 / 0.193536e6 -0.347232997e9 / 0.217728000e9 0.1788157e7 / 0.2488320e7 -0.5926349e7 / 0.37632000e8; 0.3770744693e10 / 0.3048192000e10 -0.138589901e9 / 0.62208000e8 0 0.11741773e8 / 0.5443200e7 -0.39109817e8 / 0.17418240e8 0.10216441e8 / 0.5376000e7 -0.245131109e9 / 0.217728000e9 0.92809903e8 / 0.304819200e9; -0.290431859e9 / 0.203212800e9 0.31764881e8 / 0.12441600e8 -0.11741773e8 / 0.5443200e7 0 0.4634999e7 / 0.2488320e7 -0.144219869e9 / 0.87091200e8 0.17445643e8 / 0.14515200e8 -0.38142949e8 / 0.101606400e9; 0.363704879e9 / 0.304819200e9 -0.461249e6 / 0.193536e6 0.39109817e8 / 0.17418240e8 -0.4634999e7 / 0.2488320e7 0 0.7992221e7 / 0.5443200e7 -0.817951e6 / 0.829440e6 0.4455517e7 / 0.13547520e8; -0.206906927e9 / 0.304819200e9 0.347232997e9 / 0.217728000e9 -0.10216441e8 / 0.5376000e7 0.144219869e9 / 0.87091200e8 -0.7992221e7 / 0.5443200e7 0 0.68487373e8 / 0.62208000e8 -0.1032638773e10 / 0.3048192000e10; 0.248883679e9 / 0.1016064000e10 -0.1788157e7 / 0.2488320e7 0.245131109e9 / 0.217728000e9 -0.17445643e8 / 0.14515200e8 0.817951e6 / 0.829440e6 -0.68487373e8 / 0.62208000e8 0 0.39529771e8 / 0.47628000e8; -0.2665637e7 / 0.62208000e8 0.5926349e7 / 0.37632000e8 -0.92809903e8 / 0.304819200e9 0.38142949e8 / 0.101606400e9 -0.4455517e7 / 0.13547520e8 0.1032638773e10 / 0.3048192000e10 -0.39529771e8 / 0.47628000e8 0;]; + + Q(1:8,1:8)=Q_U; + Q(m-7:m,m-7:m)=rot90( -Q_U(1:8,1:8) ,2 ); + + D1=HI*Q; + + + M_U =[0.27667249117e11 / 0.18289152000e11 -0.17100791927e11 / 0.6096384000e10 0.6123596021e10 / 0.2032128000e10 -0.12420079921e11 / 0.3657830400e10 0.3352522937e10 / 0.1219276800e10 -0.3030351383e10 / 0.2032128000e10 0.8955233071e10 / 0.18289152000e11 -0.448917533e9 / 0.6096384000e10; -0.2443029521e10 / 0.870912000e9 0.3279926909e10 / 0.373248000e9 -0.150833107e9 / 0.11612160e8 0.2418903029e10 / 0.174182400e9 -0.1195687489e10 / 0.104509440e9 0.1864443097e10 / 0.290304000e9 -0.275412413e9 / 0.124416000e9 0.26267539e8 / 0.74649600e8; 0.875033123e9 / 0.290304000e9 -0.754432799e9 / 0.58060800e8 0.262316881e9 / 0.10752000e8 -0.1615952663e10 / 0.58060800e8 0.1304948581e10 / 0.58060800e8 -0.59605951e8 / 0.4608000e7 0.270029509e9 / 0.58060800e8 -0.225377137e9 / 0.290304000e9; -0.71086111e8 / 0.20901888e8 0.2425994741e10 / 0.174182400e9 -0.1622486807e10 / 0.58060800e8 0.735382895e9 / 0.20901888e8 -0.1016121419e10 / 0.34836480e8 0.190014817e9 / 0.11612160e8 -0.447155539e9 / 0.74649600e8 0.179406911e9 / 0.174182400e9; 0.480578879e9 / 0.174182400e9 -0.6018333509e10 / 0.522547200e9 0.1319413093e10 / 0.58060800e8 -0.205032463e9 / 0.6967296e7 0.551889007e9 / 0.20901888e8 -0.887809303e9 / 0.58060800e8 0.914606453e9 / 0.174182400e9 -0.67482881e8 / 0.74649600e8; -0.434493809e9 / 0.290304000e9 0.1878773977e10 / 0.290304000e9 -0.423185977e9 / 0.32256000e8 0.964538597e9 / 0.58060800e8 -0.894343447e9 / 0.58060800e8 0.345461491e9 / 0.32256000e8 -0.1288081307e10 / 0.290304000e9 0.199200163e9 / 0.290304000e9; 0.183060319e9 / 0.373248000e9 -0.276656573e9 / 0.124416000e9 0.54579137e8 / 0.11612160e8 -0.3169984837e10 / 0.522547200e9 0.184339633e9 / 0.34836480e8 -0.1289417627e10 / 0.290304000e9 0.1431981949e10 / 0.373248000e9 -0.307164061e9 / 0.174182400e9; -0.449332253e9 / 0.6096384000e10 0.1290053923e10 / 0.3657830400e10 -0.1588745239e10 / 0.2032128000e10 0.1267377593e10 / 0.1219276800e10 -0.3326650673e10 / 0.3657830400e10 0.1396036981e10 / 0.2032128000e10 -0.2150231371e10 / 0.1219276800e10 0.52544801501e11 / 0.18289152000e11;]; + + M=-(-1/560*diag(ones(m-4,1),4)+8/315*diag(ones(m-3,1),3)-1/5*diag(ones(m-2,1),2)+8/5*diag(ones(m-1,1),1)+8/5*diag(ones(m-1,1),-1)-1/5*diag(ones(m-2,1),-2)+8/315*diag(ones(m-3,1),-3)-1/560*diag(ones(m-4,1),-4)-205/72*diag(ones(m,1),0)); + M(1:8,1:8)=M_U; + + M(m-7:m,m-7:m)=rot90( M_U ,2 ); + M=M/h; + + % DS_U=[0.363e3 / 0.140e3 -7 0.21e2 / 0.2e1 -0.35e2 / 0.3e1 0.35e2 / 0.4e1 -0.21e2 / 0.5e1 0.7e1 / 0.6e1 -0.1e1 / 0.7e1;]; + % DS=zeros(m,m); + % DS(1,1:8)=DS_U; + % DS(m,m-7:m)=fliplr(DS_U); + % DS=DS/h; + + %D2=HI*(-M+DS); + + S_U=-[0.363e3 / 0.140e3 -7 0.21e2 / 0.2e1 -0.35e2 / 0.3e1 0.35e2 / 0.4e1 -0.21e2 / 0.5e1 0.7e1 / 0.6e1 -0.1e1 / 0.7e1;]/h; + S_1=zeros(1,m); + S_1(1:8)=S_U; + S_m=zeros(1,m); + S_m(m-7:m)=fliplr(-S_U); + + + D2=HI*(-M - e_1*S_1+e_m*S_m); + S_1 = S_1'; + S_m = S_m'; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/diagInd.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,17 @@ +function A = diagInd(d,n,m) + A = zeros(n,length(d)); + for i = 1:length(d) + i0 = 1; + j0 = d(i)+1; + + I = i0 + (0:(n-1))'; + J = j0 + (0:(n-1))'; + + A(:,i) = matInd2VecInd(I,J,n); + + end +end + +function I = matInd2VecInd(i,j,n) + I = i + (j-1)*n; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher2_compatible_halfvariable.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,161 @@ +% 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] = higher2_compatible_halfvariable(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 + + H=diag(ones(m,1),0);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)); + + %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=zeros(m,1);e_1(1)=1; + e_m=zeros(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=zeros(1,m); + S_1(1:3)=S_U; + S_m=zeros(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)); + + %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)=flipud( fliplr( -Q3_U ) ); + Q3=Q3/h^2; + + + + S2_U=[1 -2 1;]/h^2; + S2_1=zeros(1,m); + S2_1(1:3)=S2_U; + S2_m=zeros(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); + + %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)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher4.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,154 @@ +function [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] = higher4(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 + + H=diag(ones(m,1),0); + H_U=[0.35809e5 / 0.100800e6 0 0 0 0 0; 0 0.13297e5 / 0.11200e5 0 0 0 0; 0 0 0.5701e4 / 0.5600e4 0 0 0; 0 0 0 0.45109e5 / 0.50400e5 0 0; 0 0 0 0 0.35191e5 / 0.33600e5 0; 0 0 0 0 0 0.33503e5 / 0.33600e5;]; + + H(1:6,1:6)=H_U; + H(m-5:m,m-5:m)=fliplr(flipud(H_U)); + H=H*h; + HI=inv(H); + + + % First derivative SBP operator, 1st order accurate at first 6 boundary points + + q2=-1/12;q1=8/12; + Q=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)); + + %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 0.526249e6 / 0.907200e6 -0.10819e5 / 0.777600e6 -0.50767e5 / 0.907200e6 -0.631e3 / 0.28800e5 0.91e2 / 0.7776e4; -0.526249e6 / 0.907200e6 0 0.1421209e7 / 0.2721600e7 0.16657e5 / 0.201600e6 -0.8467e4 / 0.453600e6 -0.33059e5 / 0.5443200e7; 0.10819e5 / 0.777600e6 -0.1421209e7 / 0.2721600e7 0 0.631187e6 / 0.1360800e7 0.400139e6 / 0.5443200e7 -0.8789e4 / 0.302400e6; 0.50767e5 / 0.907200e6 -0.16657e5 / 0.201600e6 -0.631187e6 / 0.1360800e7 0 0.496403e6 / 0.907200e6 -0.308533e6 / 0.5443200e7; 0.631e3 / 0.28800e5 0.8467e4 / 0.453600e6 -0.400139e6 / 0.5443200e7 -0.496403e6 / 0.907200e6 0 0.1805647e7 / 0.2721600e7; -0.91e2 / 0.7776e4 0.33059e5 / 0.5443200e7 0.8789e4 / 0.302400e6 0.308533e6 / 0.5443200e7 -0.1805647e7 / 0.2721600e7 0;]; + Q(1:6,1:6)=Q_U; + Q(m-5:m,m-5:m)=flipud( fliplr( -Q_U ) ); + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(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 6 boundary points + m2=1/12;m1=-16/12;m0=30/12; + M=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); + %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=[0.2386127e7 / 0.2177280e7 -0.515449e6 / 0.453600e6 -0.10781e5 / 0.777600e6 0.61567e5 / 0.1360800e7 0.6817e4 / 0.403200e6 -0.1069e4 / 0.136080e6; -0.515449e6 / 0.453600e6 0.4756039e7 / 0.2177280e7 -0.1270009e7 / 0.1360800e7 -0.3751e4 / 0.28800e5 0.3067e4 / 0.680400e6 0.119459e6 / 0.10886400e8; -0.10781e5 / 0.777600e6 -0.1270009e7 / 0.1360800e7 0.111623e6 / 0.60480e5 -0.555587e6 / 0.680400e6 -0.551339e6 / 0.5443200e7 0.8789e4 / 0.453600e6; 0.61567e5 / 0.1360800e7 -0.3751e4 / 0.28800e5 -0.555587e6 / 0.680400e6 0.1025327e7 / 0.544320e6 -0.464003e6 / 0.453600e6 0.222133e6 / 0.5443200e7; 0.6817e4 / 0.403200e6 0.3067e4 / 0.680400e6 -0.551339e6 / 0.5443200e7 -0.464003e6 / 0.453600e6 0.5074159e7 / 0.2177280e7 -0.1784047e7 / 0.1360800e7; -0.1069e4 / 0.136080e6 0.119459e6 / 0.10886400e8 0.8789e4 / 0.453600e6 0.222133e6 / 0.5443200e7 -0.1784047e7 / 0.1360800e7 0.1812749e7 / 0.725760e6;]; + + M(1:6,1:6)=M_U; + + M(m-5:m,m-5:m)=flipud( fliplr( M_U ) ); + M=M/h; + + S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; + S_1=zeros(1,m); + S_1(1:4)=S_U; + S_m=zeros(1,m); + + S_m(m-3:m)=fliplr(-S_U); + + D2=HI*(-M-e_1*S_1+e_m*S_m); + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Third derivative, 1st order accurate at first 6 boundary points + + q3=-1/8;q2=1;q1=-13/8; + Q3=q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+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)); + + %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 -0.88471e5 / 0.67200e5 0.58139e5 / 0.33600e5 -0.1167e4 / 0.2800e4 -0.89e2 / 0.11200e5 0.7e1 / 0.640e3; 0.88471e5 / 0.67200e5 0 -0.43723e5 / 0.16800e5 0.46783e5 / 0.33600e5 -0.191e3 / 0.3200e4 -0.1567e4 / 0.33600e5; -0.58139e5 / 0.33600e5 0.43723e5 / 0.16800e5 0 -0.4049e4 / 0.2400e4 0.29083e5 / 0.33600e5 -0.71e2 / 0.1400e4; 0.1167e4 / 0.2800e4 -0.46783e5 / 0.33600e5 0.4049e4 / 0.2400e4 0 -0.8591e4 / 0.5600e4 0.10613e5 / 0.11200e5; 0.89e2 / 0.11200e5 0.191e3 / 0.3200e4 -0.29083e5 / 0.33600e5 0.8591e4 / 0.5600e4 0 -0.108271e6 / 0.67200e5; -0.7e1 / 0.640e3 0.1567e4 / 0.33600e5 0.71e2 / 0.1400e4 -0.10613e5 / 0.11200e5 0.108271e6 / 0.67200e5 0;]; + + Q3(1:6,1:6)=Q3_U; + Q3(m-5:m,m-5:m)=flipud( fliplr( -Q3_U ) ); + Q3=Q3/h^2; + + + + S2_U=[2 -5 4 -1;]/h^2; + S2_1=zeros(1,m); + S2_1(1:4)=S2_U; + S2_m=zeros(1,m); + S2_m(m-3:m)=fliplr(S2_U); + + + + 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 + + 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); + + %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=[0.4596181e7 / 0.1814400e7 -0.10307743e8 / 0.1814400e7 0.160961e6 / 0.43200e5 -0.535019e6 / 0.907200e6 0.109057e6 / 0.1814400e7 -0.29273e5 / 0.604800e6; -0.10307743e8 / 0.1814400e7 0.8368543e7 / 0.604800e6 -0.9558943e7 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.11351e5 / 0.86400e5 0.204257e6 / 0.1814400e7; 0.160961e6 / 0.43200e5 -0.9558943e7 / 0.907200e6 0.4938581e7 / 0.453600e6 -0.786473e6 / 0.151200e6 0.1141057e7 / 0.907200e6 -0.120619e6 / 0.907200e6; -0.535019e6 / 0.907200e6 0.2177057e7 / 0.907200e6 -0.786473e6 / 0.151200e6 0.3146581e7 / 0.453600e6 -0.4614143e7 / 0.907200e6 0.24587e5 / 0.14400e5; 0.109057e6 / 0.1814400e7 -0.11351e5 / 0.86400e5 0.1141057e7 / 0.907200e6 -0.4614143e7 / 0.907200e6 0.185709e6 / 0.22400e5 -0.11293343e8 / 0.1814400e7; -0.29273e5 / 0.604800e6 0.204257e6 / 0.1814400e7 -0.120619e6 / 0.907200e6 0.24587e5 / 0.14400e5 -0.11293343e8 / 0.1814400e7 0.16787381e8 / 0.1814400e7;]; + + M4(1:6,1:6)=M4_U; + + M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(1,m); + S3_m(m-3:m)=fliplr(-S3_U); + + D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + + + % L=h*(m-1); + % + % x1=linspace(0,L,m)'; + % x2=x1.^2/fac(2); + % x3=x1.^3/fac(3); + % x4=x1.^4/fac(4); + % x5=x1.^5/fac(5); + % + % x0=x1.^0/fac(1); + + S_1 = S_1'; + S2_1 = S2_1'; + S3_1 = S3_1'; + S_m = S_m'; + S2_m = S2_m'; + S3_m = S3_m'; + + + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher4_compatible_halfvariable.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,154 @@ +function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher4_compatible_halfvariable(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; + + c=ones(m,1); + + + H=diag(ones(m,1),0); + H(1:4,1:4)=diag([17/48 59/48 43/48 49/48]); + H(m-3:m,m-3:m)=fliplr(flipud(diag([17/48 59/48 43/48 49/48]))); + 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)); + 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)=flipud( fliplr(-Q_U(1:4,1:4) ) ); + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(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(1:4,1:4)=M_U; + + M(m-3:m,m-3:m)=flipud( fliplr( M_U ) ); + M=M/h; + + S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; + S_1=zeros(1,m); + S_1(1:4)=S_U; + S_m=zeros(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=zeros(1,m); + S2_1(1:4)=S2_U; + S2_m=zeros(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); + + %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)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher6.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,150 @@ +function [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] = higher6(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 + + H=diag(ones(m,1),0); + H_U=[0.318365e6 / 0.1016064e7 0 0 0 0 0 0 0; 0 0.145979e6 / 0.103680e6 0 0 0 0 0 0; 0 0 0.139177e6 / 0.241920e6 0 0 0 0 0; 0 0 0 0.964969e6 / 0.725760e6 0 0 0 0; 0 0 0 0 0.593477e6 / 0.725760e6 0 0 0; 0 0 0 0 0 0.52009e5 / 0.48384e5 0 0; 0 0 0 0 0 0 0.141893e6 / 0.145152e6 0; 0 0 0 0 0 0 0 0.1019713e7 / 0.1016064e7;]; + + H(1:8,1:8)=H_U; + H(m-7:m,m-7:m)=fliplr(flipud(H_U)); + H=H*h; + HI=inv(H); + + + % First derivative SBP operator, 1st order accurate at first 6 boundary points + + q3=1/60;q2=-3/20;q1=3/4; + Q=q3*(diag(ones(m-3,1),3) - diag(ones(m-3,1),-3))+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)); + + Q_U = [0 0.1547358409e10 / 0.2421619200e10 -0.422423e6 / 0.11211200e8 -0.1002751721e10 / 0.8717829120e10 -0.15605263e8 / 0.484323840e9 0.1023865e7 / 0.24216192e8 0.291943739e9 / 0.21794572800e11 -0.24659e5 / 0.2534400e7; -0.1547358409e10 / 0.2421619200e10 0 0.23031829e8 / 0.62899200e8 0.10784027e8 / 0.34594560e8 0.2859215e7 / 0.31135104e8 -0.45982103e8 / 0.345945600e9 -0.26681e5 / 0.1182720e7 0.538846039e9 / 0.21794572800e11; 0.422423e6 / 0.11211200e8 -0.23031829e8 / 0.62899200e8 0 0.28368209e8 / 0.69189120e8 -0.9693137e7 / 0.69189120e8 0.1289363e7 / 0.17740800e8 -0.39181e5 / 0.5491200e7 -0.168647e6 / 0.24216192e8; 0.1002751721e10 / 0.8717829120e10 -0.10784027e8 / 0.34594560e8 -0.28368209e8 / 0.69189120e8 0 0.5833151e7 / 0.10644480e8 0.4353179e7 / 0.69189120e8 0.2462459e7 / 0.155675520e9 -0.215471e6 / 0.10762752e8; 0.15605263e8 / 0.484323840e9 -0.2859215e7 / 0.31135104e8 0.9693137e7 / 0.69189120e8 -0.5833151e7 / 0.10644480e8 0 0.7521509e7 / 0.13837824e8 -0.1013231e7 / 0.11531520e8 0.103152839e9 / 0.8717829120e10; -0.1023865e7 / 0.24216192e8 0.45982103e8 / 0.345945600e9 -0.1289363e7 / 0.17740800e8 -0.4353179e7 / 0.69189120e8 -0.7521509e7 / 0.13837824e8 0 0.67795697e8 / 0.98841600e8 -0.17263733e8 / 0.151351200e9; -0.291943739e9 / 0.21794572800e11 0.26681e5 / 0.1182720e7 0.39181e5 / 0.5491200e7 -0.2462459e7 / 0.155675520e9 0.1013231e7 / 0.11531520e8 -0.67795697e8 / 0.98841600e8 0 0.1769933569e10 / 0.2421619200e10; 0.24659e5 / 0.2534400e7 -0.538846039e9 / 0.21794572800e11 0.168647e6 / 0.24216192e8 0.215471e6 / 0.10762752e8 -0.103152839e9 / 0.8717829120e10 0.17263733e8 / 0.151351200e9 -0.1769933569e10 / 0.2421619200e10 0;]; + + Q(1:8,1:8)=Q_U; + Q(m-7:m,m-7:m)=flipud( fliplr( -Q_U ) ); + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(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 6 boundary points + m3=-1/90;m2=3/20;m1=-3/2;m0=49/18; + + M=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); + M_U=[0.4347276223e10 / 0.3736212480e10 -0.1534657609e10 / 0.1210809600e10 0.68879e5 / 0.3057600e7 0.1092927401e10 / 0.13076743680e11 0.18145423e8 / 0.968647680e9 -0.1143817e7 / 0.60540480e8 -0.355447739e9 / 0.65383718400e11 0.56081e5 / 0.16473600e8; -0.1534657609e10 / 0.1210809600e10 0.42416226217e11 / 0.18681062400e11 -0.228654119e9 / 0.345945600e9 -0.12245627e8 / 0.34594560e8 -0.2995295e7 / 0.46702656e8 0.52836503e8 / 0.691891200e9 0.119351e6 / 0.12812800e8 -0.634102039e9 / 0.65383718400e11; 0.68879e5 / 0.3057600e7 -0.228654119e9 / 0.345945600e9 0.5399287e7 / 0.4193280e7 -0.24739409e8 / 0.34594560e8 0.7878737e7 / 0.69189120e8 -0.1917829e7 / 0.31449600e8 0.39727e5 / 0.3660800e7 0.10259e5 / 0.4656960e7; 0.1092927401e10 / 0.13076743680e11 -0.12245627e8 / 0.34594560e8 -0.24739409e8 / 0.34594560e8 0.7780367599e10 / 0.3736212480e10 -0.70085363e8 / 0.69189120e8 -0.500209e6 / 0.6289920e7 -0.311543e6 / 0.17962560e8 0.278191e6 / 0.21525504e8; 0.18145423e8 / 0.968647680e9 -0.2995295e7 / 0.46702656e8 0.7878737e7 / 0.69189120e8 -0.70085363e8 / 0.69189120e8 0.7116321131e10 / 0.3736212480e10 -0.545081e6 / 0.532224e6 0.811631e6 / 0.11531520e8 -0.84101639e8 / 0.13076743680e11; -0.1143817e7 / 0.60540480e8 0.52836503e8 / 0.691891200e9 -0.1917829e7 / 0.31449600e8 -0.500209e6 / 0.6289920e7 -0.545081e6 / 0.532224e6 0.324760747e9 / 0.138378240e9 -0.65995697e8 / 0.49420800e8 0.1469203e7 / 0.13759200e8; -0.355447739e9 / 0.65383718400e11 0.119351e6 / 0.12812800e8 0.39727e5 / 0.3660800e7 -0.311543e6 / 0.17962560e8 0.811631e6 / 0.11531520e8 -0.65995697e8 / 0.49420800e8 0.48284442317e11 / 0.18681062400e11 -0.1762877569e10 / 0.1210809600e10; 0.56081e5 / 0.16473600e8 -0.634102039e9 / 0.65383718400e11 0.10259e5 / 0.4656960e7 0.278191e6 / 0.21525504e8 -0.84101639e8 / 0.13076743680e11 0.1469203e7 / 0.13759200e8 -0.1762877569e10 / 0.1210809600e10 0.10117212851e11 / 0.3736212480e10;]; + + M(1:8,1:8)=M_U; + + M(m-7:m,m-7:m)=flipud( fliplr( M_U ) ); + M=M/h; + + S_U=[-0.25e2 / 0.12e2 4 -3 0.4e1 / 0.3e1 -0.1e1 / 0.4e1;]/h; + S_1=zeros(1,m); + S_1(1:5)=S_U; + S_m=zeros(1,m); + + S_m(m-4:m)=fliplr(-S_U); + + D2=HI*(-M-e_1*S_1+e_m*S_m); + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Third derivative, 1st order accurate at first 6 boundary points + + q4=7/240;q3=-3/10;q2=169/120;q1=-61/30; + Q3=q4*(diag(ones(m-4,1),4)-diag(ones(m-4,1),-4))+q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+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)); + + %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 -0.10882810591e11 / 0.5811886080e10 0.398713069e9 / 0.132088320e9 -0.1746657571e10 / 0.1162377216e10 0.56050639e8 / 0.145297152e9 -0.11473393e8 / 0.1162377216e10 -0.38062741e8 / 0.1452971520e10 0.30473e5 / 0.4392960e7; 0.10882810591e11 / 0.5811886080e10 0 -0.3720544343e10 / 0.830269440e9 0.767707019e9 / 0.207567360e9 -0.1047978301e10 / 0.830269440e9 0.1240729e7 / 0.14826240e8 0.6807397e7 / 0.55351296e8 -0.50022767e8 / 0.1452971520e10; -0.398713069e9 / 0.132088320e9 0.3720544343e10 / 0.830269440e9 0 -0.2870078009e10 / 0.830269440e9 0.74962049e8 / 0.29652480e8 -0.12944857e8 / 0.30750720e8 -0.17846623e8 / 0.103783680e9 0.68707591e8 / 0.1162377216e10; 0.1746657571e10 / 0.1162377216e10 -0.767707019e9 / 0.207567360e9 0.2870078009e10 / 0.830269440e9 0 -0.727867087e9 / 0.276756480e9 0.327603877e9 / 0.207567360e9 -0.175223717e9 / 0.830269440e9 0.1353613e7 / 0.726485760e9; -0.56050639e8 / 0.145297152e9 0.1047978301e10 / 0.830269440e9 -0.74962049e8 / 0.29652480e8 0.727867087e9 / 0.276756480e9 0 -0.1804641793e10 / 0.830269440e9 0.311038417e9 / 0.207567360e9 -0.1932566239e10 / 0.5811886080e10; 0.11473393e8 / 0.1162377216e10 -0.1240729e7 / 0.14826240e8 0.12944857e8 / 0.30750720e8 -0.327603877e9 / 0.207567360e9 0.1804641793e10 / 0.830269440e9 0 -0.1760949511e10 / 0.830269440e9 0.2105883973e10 / 0.1452971520e10; 0.38062741e8 / 0.1452971520e10 -0.6807397e7 / 0.55351296e8 0.17846623e8 / 0.103783680e9 0.175223717e9 / 0.830269440e9 -0.311038417e9 / 0.207567360e9 0.1760949511e10 / 0.830269440e9 0 -0.1081094773e10 / 0.528353280e9; -0.30473e5 / 0.4392960e7 0.50022767e8 / 0.1452971520e10 -0.68707591e8 / 0.1162377216e10 -0.1353613e7 / 0.726485760e9 0.1932566239e10 / 0.5811886080e10 -0.2105883973e10 / 0.1452971520e10 0.1081094773e10 / 0.528353280e9 0;]; + + Q3(1:8,1:8)=Q3_U; + Q3(m-7:m,m-7:m)=flipud( fliplr( -Q3_U ) ); + Q3=Q3/h^2; + + + + 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=zeros(1,m); + S2_1(1:5)=S2_U; + S2_m=zeros(1,m); + S2_m(m-4:m)=fliplr(S2_U); + + + + 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 + + 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); + + %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=[0.40833734273e11 / 0.10761070320e11 -0.162181998421e12 / 0.16397821440e11 0.4696168417e10 / 0.521748864e9 -0.245714671483e12 / 0.68870850048e11 0.2185939219e10 / 0.2869618752e10 -0.15248255797e11 / 0.114784750080e12 0.345156907e9 / 0.12298366080e11 0.6388381e7 / 0.1093188096e10; -0.162181998421e12 / 0.16397821440e11 0.147281127041e12 / 0.5380535160e10 -0.3072614435609e13 / 0.114784750080e12 0.320122985851e12 / 0.28696187520e11 -0.768046031383e12 / 0.344354250240e12 0.7861605187e10 / 0.14348093760e11 -0.803762437e9 / 0.4251287040e10 0.167394281e9 / 0.86088562560e11; 0.4696168417e10 / 0.521748864e9 -0.3072614435609e13 / 0.114784750080e12 0.139712483333e12 / 0.4782697920e10 -0.1634124842747e13 / 0.114784750080e12 0.90855193447e11 / 0.28696187520e11 -0.26412188989e11 / 0.38261583360e11 0.668741173e9 / 0.1793511720e10 -0.132673781e9 / 0.2342545920e10; -0.245714671483e12 / 0.68870850048e11 0.320122985851e12 / 0.28696187520e11 -0.1634124842747e13 / 0.114784750080e12 0.437353997177e12 / 0.43044281280e11 -0.172873969321e12 / 0.38261583360e11 0.34759553483e11 / 0.28696187520e11 -0.98928859751e11 / 0.344354250240e12 0.295000207e9 / 0.3587023440e10; 0.2185939219e10 / 0.2869618752e10 -0.768046031383e12 / 0.344354250240e12 0.90855193447e11 / 0.28696187520e11 -0.172873969321e12 / 0.38261583360e11 0.126711914423e12 / 0.21522140640e11 -0.520477408939e12 / 0.114784750080e12 0.49581230003e11 / 0.28696187520e11 -0.99640101991e11 / 0.344354250240e12; -0.15248255797e11 / 0.114784750080e12 0.7861605187e10 / 0.14348093760e11 -0.26412188989e11 / 0.38261583360e11 0.34759553483e11 / 0.28696187520e11 -0.520477408939e12 / 0.114784750080e12 0.19422074929e11 / 0.2391348960e10 -0.772894368601e12 / 0.114784750080e12 0.10579712849e11 / 0.4099455360e10; 0.345156907e9 / 0.12298366080e11 -0.803762437e9 / 0.4251287040e10 0.668741173e9 / 0.1793511720e10 -0.98928859751e11 / 0.344354250240e12 0.49581230003e11 / 0.28696187520e11 -0.772894368601e12 / 0.114784750080e12 0.456715296239e12 / 0.43044281280e11 -0.915425403107e12 / 0.114784750080e12; 0.6388381e7 / 0.1093188096e10 0.167394281e9 / 0.86088562560e11 -0.132673781e9 / 0.2342545920e10 0.295000207e9 / 0.3587023440e10 -0.99640101991e11 / 0.344354250240e12 0.10579712849e11 / 0.4099455360e10 -0.915425403107e12 / 0.114784750080e12 0.488029542379e12 / 0.43044281280e11;]; + + M4(1:8,1:8)=M4_U; + + M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:5)=S3_U; + S3_m=zeros(1,m); + S3_m(m-4:m)=fliplr(-S3_U); + + D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + + + L=h*(m-1); + + % x1=linspace(0,L,m)'; + % x2=x1.^2/fac(2); + % x3=x1.^3/fac(3); + % x4=x1.^4/fac(4); + % x5=x1.^5/fac(5); + + % x0=x1.^0/fac(1); + + S_1 = S_1'; + S2_1 = S2_1'; + S3_1 = S3_1'; + S_m = S_m'; + S2_m = S2_m'; + S3_m = S3_m'; + + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher6_compatible_halfvariable.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,160 @@ +function [H, HI, D2, D4, e_1, e_m, M4, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher6_compatible_halfvariable(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; + + + H=diag(ones(m,1),0); + 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)=fliplr(flipud(diag([13649/43200,12013/8640, ... + 2711/4320,5359/4320,7877/8640,43801/43200]))); + + + 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)=flipud( fliplr(-D1(1:6,1:9))); + D1=D1/h; + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + S_U=[-25/12, 4, -3, 4/3, -1/4]/h; + S_1=zeros(1,m); + S_1(1:5)=S_U; + S_m=zeros(1,m); + S_m(m-4:m)=fliplr(-S_U); + S_1 = S_1'; + S_m = S_m'; + + + + %DS=zeros(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); 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-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=zeros(1,m); + S2_1(1:5)=S2_U; + S2_m=zeros(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); + + %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)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:5)=S3_U; + S3_m=zeros(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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_compatible2.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,143 @@ +function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_compatible2(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 + + H=diag(ones(m,1),0);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)); + + %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=zeros(m,1);e_1(1)=1; + e_m=zeros(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 6 boundary points + 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; + + S_U=[-1 1]/h; + S_1=zeros(1,m); + S_1(1:2)=S_U; + S_m=zeros(1,m); + + S_m(m-1:m)=fliplr(-S_U); + + D2=HI*(-M-e_1*S_1+e_m*S_m); + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % 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)); + + %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 -0.2e1 / 0.5e1 0.3e1 / 0.10e2 0.1e1 / 0.10e2; 0.2e1 / 0.5e1 0 -0.7e1 / 0.10e2 0.3e1 / 0.10e2; -0.3e1 / 0.10e2 0.7e1 / 0.10e2 0 -0.9e1 / 0.10e2; -0.1e1 / 0.10e2 -0.3e1 / 0.10e2 0.9e1 / 0.10e2 0;]; + Q3(1:4,1:4)=Q3_U; + Q3(m-3:m,m-3:m)=flipud( fliplr( -Q3_U ) ); + Q3=Q3/h^2; + + + + S2_U=[1 -2 1;]/h^2; + S2_1=zeros(1,m); + S2_1(1:3)=S2_U; + S2_m=zeros(1,m); + S2_m(m-2:m)=fliplr(S2_U); + + + + 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); + + %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=[0.4e1 / 0.5e1 -0.7e1 / 0.5e1 0.2e1 / 0.5e1 0.1e1 / 0.5e1; -0.7e1 / 0.5e1 0.16e2 / 0.5e1 -0.11e2 / 0.5e1 0.2e1 / 0.5e1; 0.2e1 / 0.5e1 -0.11e2 / 0.5e1 0.21e2 / 0.5e1 -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)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(1,m); + S3_m(m-3:m)=fliplr(-S3_U); + + D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + + + + S_1 = S_1'; + S_m = S_m'; + S2_1 = S2_1'; + S2_m = S2_m'; + S3_1 = S3_1'; + S3_m = S3_m'; + + + + + % L=h*(m-1); + + % x1=linspace(0,L,m)'; + % x2=x1.^2/fac(2); + % x3=x1.^3/fac(3); + % x4=x1.^4/fac(4); + % x5=x1.^5/fac(5); + + % x0=x1.^0/fac(1); + + +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_compatible4.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,148 @@ +function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_compatible4(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 + + H=diag(ones(m,1),0); + H_U=[0.3e1 / 0.11e2 0 0 0 0 0; 0 0.2125516311e10 / 0.1311004640e10 0 0 0 0; 0 0 0.278735189e9 / 0.1966506960e10 0 0 0; 0 0 0 0.285925927e9 / 0.163875580e9 0 0; 0 0 0 0 0.1284335339e10 / 0.1966506960e10 0; 0 0 0 0 0 0.4194024163e10 / 0.3933013920e10;]; + H(1:6,1:6)=H_U; + H(m-5:m,m-5:m)=fliplr(flipud(H_U)); + H=H*h; + HI=inv(H); + + + % First derivative SBP operator, 1st order accurate at first 6 boundary points + + q2=-1/12;q1=8/12; + Q=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)); + + %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 0.9e1 / 0.11e2 -0.9e1 / 0.22e2 0.1e1 / 0.11e2 0 0; -0.9e1 / 0.11e2 0 0.2595224893e10 / 0.2622009280e10 -0.151435707e9 / 0.327751160e9 0.1112665611e10 / 0.2622009280e10 -0.1290899e7 / 0.9639740e7; 0.9e1 / 0.22e2 -0.2595224893e10 / 0.2622009280e10 0 0.1468436423e10 / 0.983253480e9 -0.1194603401e10 / 0.983253480e9 0.72033031e8 / 0.238364480e9; -0.1e1 / 0.11e2 0.151435707e9 / 0.327751160e9 -0.1468436423e10 / 0.983253480e9 0 0.439819541e9 / 0.327751160e9 -0.215942641e9 / 0.983253480e9; 0 -0.1112665611e10 / 0.2622009280e10 0.1194603401e10 / 0.983253480e9 -0.439819541e9 / 0.327751160e9 0 0.1664113643e10 / 0.2622009280e10; 0 0.1290899e7 / 0.9639740e7 -0.72033031e8 / 0.238364480e9 0.215942641e9 / 0.983253480e9 -0.1664113643e10 / 0.2622009280e10 0;]; + Q(1:6,1:6)=Q_U; + Q(m-5:m,m-5:m)=flipud( fliplr( -Q_U ) ); + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(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 6 boundary points + % m2=1/12;m1=-16/12;m0=30/12; + % M=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); + % %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=[0.2386127e7 / 0.2177280e7 -0.515449e6 / 0.453600e6 -0.10781e5 / 0.777600e6 0.61567e5 / 0.1360800e7 0.6817e4 / 0.403200e6 -0.1069e4 / 0.136080e6; -0.515449e6 / 0.453600e6 0.4756039e7 / 0.2177280e7 -0.1270009e7 / 0.1360800e7 -0.3751e4 / 0.28800e5 0.3067e4 / 0.680400e6 0.119459e6 / 0.10886400e8; -0.10781e5 / 0.777600e6 -0.1270009e7 / 0.1360800e7 0.111623e6 / 0.60480e5 -0.555587e6 / 0.680400e6 -0.551339e6 / 0.5443200e7 0.8789e4 / 0.453600e6; 0.61567e5 / 0.1360800e7 -0.3751e4 / 0.28800e5 -0.555587e6 / 0.680400e6 0.1025327e7 / 0.544320e6 -0.464003e6 / 0.453600e6 0.222133e6 / 0.5443200e7; 0.6817e4 / 0.403200e6 0.3067e4 / 0.680400e6 -0.551339e6 / 0.5443200e7 -0.464003e6 / 0.453600e6 0.5074159e7 / 0.2177280e7 -0.1784047e7 / 0.1360800e7; -0.1069e4 / 0.136080e6 0.119459e6 / 0.10886400e8 0.8789e4 / 0.453600e6 0.222133e6 / 0.5443200e7 -0.1784047e7 / 0.1360800e7 0.1812749e7 / 0.725760e6;]; + % + % M(1:6,1:6)=M_U; + % + % M(m-5:m,m-5:m)=flipud( fliplr( M_U ) ); + % M=M/h; + % + S_U=[-0.11e2 / 0.6e1 3 -0.3e1 / 0.2e1 0.1e1 / 0.3e1;]/h; + S_1=zeros(1,m); + S_1(1:4)=S_U; + S_m=zeros(1,m); + + S_m(m-3:m)=fliplr(-S_U); + + % D2=HI*(-M-e_1*S_1+e_m*S_m); + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Third derivative, 1st order accurate at first 6 boundary points + + % q3=-1/8;q2=1;q1=-13/8; + % Q3=q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+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)); + % + % %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 -0.88471e5 / 0.67200e5 0.58139e5 / 0.33600e5 -0.1167e4 / 0.2800e4 -0.89e2 / 0.11200e5 0.7e1 / 0.640e3; 0.88471e5 / 0.67200e5 0 -0.43723e5 / 0.16800e5 0.46783e5 / 0.33600e5 -0.191e3 / 0.3200e4 -0.1567e4 / 0.33600e5; -0.58139e5 / 0.33600e5 0.43723e5 / 0.16800e5 0 -0.4049e4 / 0.2400e4 0.29083e5 / 0.33600e5 -0.71e2 / 0.1400e4; 0.1167e4 / 0.2800e4 -0.46783e5 / 0.33600e5 0.4049e4 / 0.2400e4 0 -0.8591e4 / 0.5600e4 0.10613e5 / 0.11200e5; 0.89e2 / 0.11200e5 0.191e3 / 0.3200e4 -0.29083e5 / 0.33600e5 0.8591e4 / 0.5600e4 0 -0.108271e6 / 0.67200e5; -0.7e1 / 0.640e3 0.1567e4 / 0.33600e5 0.71e2 / 0.1400e4 -0.10613e5 / 0.11200e5 0.108271e6 / 0.67200e5 0;]; + % + % Q3(1:6,1:6)=Q3_U; + % Q3(m-5:m,m-5:m)=flipud( fliplr( -Q3_U ) ); + % Q3=Q3/h^2; + + + + S2_U=[2 -5 4 -1;]/h^2; + S2_1=zeros(1,m); + S2_1(1:4)=S2_U; + S2_m=zeros(1,m); + S2_m(m-3:m)=fliplr(S2_U); + + + + %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 + + 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); + + %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=[0.227176919517319e15 / 0.94899692875680e14 -0.15262605263734e14 / 0.2965615402365e13 0.20205404771243e14 / 0.6778549491120e13 -0.3998303664097e13 / 0.23724923218920e14 0.1088305091927e13 / 0.94899692875680e14 -0.1686077077693e13 / 0.23724923218920e14; -0.15262605263734e14 / 0.2965615402365e13 0.280494781164181e15 / 0.23724923218920e14 -0.46417445546261e14 / 0.5931230804730e13 0.1705307929429e13 / 0.1694637372780e13 -0.553547394061e12 / 0.5931230804730e13 0.5615721694973e13 / 0.23724923218920e14; 0.20205404771243e14 / 0.6778549491120e13 -0.46417445546261e14 / 0.5931230804730e13 0.4135802350237e13 / 0.551742400440e12 -0.4140981465247e13 / 0.1078405600860e13 0.75538453067437e14 / 0.47449846437840e14 -0.4778134936391e13 / 0.11862461609460e14; -0.3998303664097e13 / 0.23724923218920e14 0.1705307929429e13 / 0.1694637372780e13 -0.4140981465247e13 / 0.1078405600860e13 0.20760974175677e14 / 0.2965615402365e13 -0.138330689701889e15 / 0.23724923218920e14 0.23711317526909e14 / 0.11862461609460e14; 0.1088305091927e13 / 0.94899692875680e14 -0.553547394061e12 / 0.5931230804730e13 0.75538453067437e14 / 0.47449846437840e14 -0.138330689701889e15 / 0.23724923218920e14 0.120223780251937e15 / 0.13557098982240e14 -0.151383731537477e15 / 0.23724923218920e14; -0.1686077077693e13 / 0.23724923218920e14 0.5615721694973e13 / 0.23724923218920e14 -0.4778134936391e13 / 0.11862461609460e14 0.23711317526909e14 / 0.11862461609460e14 -0.151383731537477e15 / 0.23724923218920e14 0.220304030094121e15 / 0.23724923218920e14;]; + + M4(1:6,1:6)=M4_U; + + M4(m-5:m,m-5:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-1 3 -3 1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:4)=S3_U; + S3_m=zeros(1,m); + S3_m(m-3:m)=fliplr(-S3_U); + + D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + + S_1 = S_1'; + S_m = S_m'; + S2_1 = S2_1'; + S2_m = S2_m'; + S3_1 = S3_1'; + S3_m = S3_m'; + + % L=h*(m-1); + % + % x1=linspace(0,L,m)'; + % x2=x1.^2/fac(2); + % x3=x1.^3/fac(3); + % x4=x1.^4/fac(4); + % x5=x1.^5/fac(5); + % + % x0=x1.^0/fac(1); + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/higher_compatible6.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,148 @@ +function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m, S3_1, S3_m, S_1, S_m] = higher_compatible6(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 + + H=diag(ones(m,1),0); + H_U=[0.7493827e7 / 0.25401600e8 0 0 0 0 0 0 0; 0 0.5534051e7 / 0.3628800e7 0 0 0 0 0 0; 0 0 0.104561e6 / 0.403200e6 0 0 0 0 0; 0 0 0 0.260503e6 / 0.145152e6 0 0 0 0; 0 0 0 0 0.43237e5 / 0.103680e6 0 0 0; 0 0 0 0 0 0.514081e6 / 0.403200e6 0 0; 0 0 0 0 0 0 0.3356179e7 / 0.3628800e7 0; 0 0 0 0 0 0 0 0.25631027e8 / 0.25401600e8;]; + + H(1:8,1:8)=H_U; + H(m-7:m,m-7:m)=fliplr(flipud(H_U)); + H=H*h; + HI=inv(H); + + + % First derivative SBP operator, 3rd order accurate at first 8 boundary points + + q3=1/60;q2=-3/20;q1=3/4; + Q=q3*(diag(ones(m-3,1),3) - diag(ones(m-3,1),-3))+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)); + + Q_U = [0 0.26431903e8 / 0.43545600e8 0.39791489e8 / 0.152409600e9 -0.12747751e8 / 0.16934400e8 0.76099447e8 / 0.152409600e9 -0.1397443e7 / 0.12192768e8 0 0; -0.26431903e8 / 0.43545600e8 0 -0.13847476213e11 / 0.19559232000e11 0.35844843977e11 / 0.11735539200e11 -0.63413503537e11 / 0.23471078400e11 0.4764412871e10 / 0.3911846400e10 -0.1668252557e10 / 0.5867769600e10 0.842644697e9 / 0.29338848000e11; -0.39791489e8 / 0.152409600e9 0.13847476213e11 / 0.19559232000e11 0 -0.73834802771e11 / 0.23471078400e11 0.1802732209e10 / 0.325987200e9 -0.65514173e8 / 0.16299360e8 0.79341409141e11 / 0.58677696000e11 -0.1282384321e10 / 0.7823692800e10; 0.12747751e8 / 0.16934400e8 -0.35844843977e11 / 0.11735539200e11 0.73834802771e11 / 0.23471078400e11 0 -0.5274106087e10 / 0.1173553920e10 0.33743985841e11 / 0.5867769600e10 -0.6482602549e10 / 0.2607897600e10 0.1506017269e10 / 0.3911846400e10; -0.76099447e8 / 0.152409600e9 0.63413503537e11 / 0.23471078400e11 -0.1802732209e10 / 0.325987200e9 0.5274106087e10 / 0.1173553920e10 0 -0.7165829063e10 / 0.2607897600e10 0.23903110999e11 / 0.11735539200e11 -0.5346675911e10 / 0.11735539200e11; 0.1397443e7 / 0.12192768e8 -0.4764412871e10 / 0.3911846400e10 0.65514173e8 / 0.16299360e8 -0.33743985841e11 / 0.5867769600e10 0.7165829063e10 / 0.2607897600e10 0 -0.1060918223e10 / 0.11735539200e11 0.628353989e9 / 0.3911846400e10; 0 0.1668252557e10 / 0.5867769600e10 -0.79341409141e11 / 0.58677696000e11 0.6482602549e10 / 0.2607897600e10 -0.23903110999e11 / 0.11735539200e11 0.1060918223e10 / 0.11735539200e11 0 0.25889988599e11 / 0.39118464000e11; 0 -0.842644697e9 / 0.29338848000e11 0.1282384321e10 / 0.7823692800e10 -0.1506017269e10 / 0.3911846400e10 0.5346675911e10 / 0.11735539200e11 -0.628353989e9 / 0.3911846400e10 -0.25889988599e11 / 0.39118464000e11 0;]; + + Q(1:8,1:8)=Q_U; + Q(m-7:m,m-7:m)=flipud( fliplr( -Q_U ) ); + + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(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 6 boundary points + % m3=-1/90;m2=3/20;m1=-3/2;m0=49/18; + % + % M=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); + % M_U=[0.4347276223e10 / 0.3736212480e10 -0.1534657609e10 / 0.1210809600e10 0.68879e5 / 0.3057600e7 0.1092927401e10 / 0.13076743680e11 0.18145423e8 / 0.968647680e9 -0.1143817e7 / 0.60540480e8 -0.355447739e9 / 0.65383718400e11 0.56081e5 / 0.16473600e8; -0.1534657609e10 / 0.1210809600e10 0.42416226217e11 / 0.18681062400e11 -0.228654119e9 / 0.345945600e9 -0.12245627e8 / 0.34594560e8 -0.2995295e7 / 0.46702656e8 0.52836503e8 / 0.691891200e9 0.119351e6 / 0.12812800e8 -0.634102039e9 / 0.65383718400e11; 0.68879e5 / 0.3057600e7 -0.228654119e9 / 0.345945600e9 0.5399287e7 / 0.4193280e7 -0.24739409e8 / 0.34594560e8 0.7878737e7 / 0.69189120e8 -0.1917829e7 / 0.31449600e8 0.39727e5 / 0.3660800e7 0.10259e5 / 0.4656960e7; 0.1092927401e10 / 0.13076743680e11 -0.12245627e8 / 0.34594560e8 -0.24739409e8 / 0.34594560e8 0.7780367599e10 / 0.3736212480e10 -0.70085363e8 / 0.69189120e8 -0.500209e6 / 0.6289920e7 -0.311543e6 / 0.17962560e8 0.278191e6 / 0.21525504e8; 0.18145423e8 / 0.968647680e9 -0.2995295e7 / 0.46702656e8 0.7878737e7 / 0.69189120e8 -0.70085363e8 / 0.69189120e8 0.7116321131e10 / 0.3736212480e10 -0.545081e6 / 0.532224e6 0.811631e6 / 0.11531520e8 -0.84101639e8 / 0.13076743680e11; -0.1143817e7 / 0.60540480e8 0.52836503e8 / 0.691891200e9 -0.1917829e7 / 0.31449600e8 -0.500209e6 / 0.6289920e7 -0.545081e6 / 0.532224e6 0.324760747e9 / 0.138378240e9 -0.65995697e8 / 0.49420800e8 0.1469203e7 / 0.13759200e8; -0.355447739e9 / 0.65383718400e11 0.119351e6 / 0.12812800e8 0.39727e5 / 0.3660800e7 -0.311543e6 / 0.17962560e8 0.811631e6 / 0.11531520e8 -0.65995697e8 / 0.49420800e8 0.48284442317e11 / 0.18681062400e11 -0.1762877569e10 / 0.1210809600e10; 0.56081e5 / 0.16473600e8 -0.634102039e9 / 0.65383718400e11 0.10259e5 / 0.4656960e7 0.278191e6 / 0.21525504e8 -0.84101639e8 / 0.13076743680e11 0.1469203e7 / 0.13759200e8 -0.1762877569e10 / 0.1210809600e10 0.10117212851e11 / 0.3736212480e10;]; + % + % M(1:8,1:8)=M_U; + % + % M(m-7:m,m-7:m)=flipud( fliplr( M_U ) ); + % M=M/h; + + S_U=[-0.12700800e8 / 0.7493827e7 0.185023321e9 / 0.89925924e8 0.39791489e8 / 0.44962962e8 -0.38243253e8 / 0.14987654e8 0.76099447e8 / 0.44962962e8 -0.34936075e8 / 0.89925924e8;]/h; + S_1=zeros(1,m); + S_1(1:6)=S_U; + S_m=zeros(1,m); + + S_m(m-5:m)=fliplr(-S_U); + + %D2=HI*(-M-e_1*S_1+e_m*S_m); + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + % Third derivative, 1st order accurate at first 6 boundary points + + % q4=7/240;q3=-3/10;q2=169/120;q1=-61/30; + % Q3=q4*(diag(ones(m-4,1),4)-diag(ones(m-4,1),-4))+q3*(diag(ones(m-3,1),3)-diag(ones(m-3,1),-3))+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)); + % + % %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 -0.10882810591e11 / 0.5811886080e10 0.398713069e9 / 0.132088320e9 -0.1746657571e10 / 0.1162377216e10 0.56050639e8 / 0.145297152e9 -0.11473393e8 / 0.1162377216e10 -0.38062741e8 / 0.1452971520e10 0.30473e5 / 0.4392960e7; 0.10882810591e11 / 0.5811886080e10 0 -0.3720544343e10 / 0.830269440e9 0.767707019e9 / 0.207567360e9 -0.1047978301e10 / 0.830269440e9 0.1240729e7 / 0.14826240e8 0.6807397e7 / 0.55351296e8 -0.50022767e8 / 0.1452971520e10; -0.398713069e9 / 0.132088320e9 0.3720544343e10 / 0.830269440e9 0 -0.2870078009e10 / 0.830269440e9 0.74962049e8 / 0.29652480e8 -0.12944857e8 / 0.30750720e8 -0.17846623e8 / 0.103783680e9 0.68707591e8 / 0.1162377216e10; 0.1746657571e10 / 0.1162377216e10 -0.767707019e9 / 0.207567360e9 0.2870078009e10 / 0.830269440e9 0 -0.727867087e9 / 0.276756480e9 0.327603877e9 / 0.207567360e9 -0.175223717e9 / 0.830269440e9 0.1353613e7 / 0.726485760e9; -0.56050639e8 / 0.145297152e9 0.1047978301e10 / 0.830269440e9 -0.74962049e8 / 0.29652480e8 0.727867087e9 / 0.276756480e9 0 -0.1804641793e10 / 0.830269440e9 0.311038417e9 / 0.207567360e9 -0.1932566239e10 / 0.5811886080e10; 0.11473393e8 / 0.1162377216e10 -0.1240729e7 / 0.14826240e8 0.12944857e8 / 0.30750720e8 -0.327603877e9 / 0.207567360e9 0.1804641793e10 / 0.830269440e9 0 -0.1760949511e10 / 0.830269440e9 0.2105883973e10 / 0.1452971520e10; 0.38062741e8 / 0.1452971520e10 -0.6807397e7 / 0.55351296e8 0.17846623e8 / 0.103783680e9 0.175223717e9 / 0.830269440e9 -0.311038417e9 / 0.207567360e9 0.1760949511e10 / 0.830269440e9 0 -0.1081094773e10 / 0.528353280e9; -0.30473e5 / 0.4392960e7 0.50022767e8 / 0.1452971520e10 -0.68707591e8 / 0.1162377216e10 -0.1353613e7 / 0.726485760e9 0.1932566239e10 / 0.5811886080e10 -0.2105883973e10 / 0.1452971520e10 0.1081094773e10 / 0.528353280e9 0;]; + % + % Q3(1:8,1:8)=Q3_U; + % Q3(m-7:m,m-7:m)=flipud( fliplr( -Q3_U ) ); + % Q3=Q3/h^2; + % + % + % + 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=zeros(1,m); + S2_1(1:5)=S2_U; + S2_m=zeros(1,m); + S2_m(m-4:m)=fliplr(S2_U); + % + % + % + % 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 + + 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); + + + + M4_U=[0.600485868980522851e18 / 0.274825314114120000e18 -0.1421010223681841e16 / 0.348984525859200e15 0.38908412970187e14 / 0.1586293299360000e16 0.10224077451922837e17 / 0.2243471951952000e16 -0.7577302712815639e16 / 0.1744922629296000e16 0.138091642084013e15 / 0.59351109840000e14 -0.3775041725375197e16 / 0.4486943903904000e16 0.9907210230881393e16 / 0.61072292025360000e17; -0.1421010223681841e16 / 0.348984525859200e15 0.3985852497808703e16 / 0.407903991264000e15 -0.90048788923861e14 / 0.15579666333000e14 -0.4312795866499e13 / 0.997098645312e12 0.4414634708891947e16 / 0.448694390390400e15 -0.886174803100459e15 / 0.99709864531200e14 0.4333e4 / 0.1000e4 -0.13800578064893047e17 / 0.15704303663664000e17; 0.38908412970187e14 / 0.1586293299360000e16 -0.90048788923861e14 / 0.15579666333000e14 0.2071682582321887e16 / 0.113306664240000e15 -0.769471337294003e15 / 0.41545776888000e14 0.112191585452033e15 / 0.166183107552000e15 0.7204491902193671e16 / 0.623186653320000e15 -0.24847093554379e14 / 0.3115933266600e13 0.943854037768721e15 / 0.545288321655000e15; 0.10224077451922837e17 / 0.2243471951952000e16 -0.4312795866499e13 / 0.997098645312e12 -0.769471337294003e15 / 0.41545776888000e14 0.3086874339649421e16 / 0.81580798252800e14 -0.396009005312111e15 / 0.16618310755200e14 0.348854811893087e15 / 0.249274661328000e15 0.895954627955053e15 / 0.224347195195200e15 -0.184881685054543e15 / 0.166183107552000e15; -0.7577302712815639e16 / 0.1744922629296000e16 0.4414634708891947e16 / 0.448694390390400e15 0.112191585452033e15 / 0.166183107552000e15 -0.396009005312111e15 / 0.16618310755200e14 0.3774861828677557e16 / 0.112173597597600e15 -0.5693689108983593e16 / 0.249274661328000e15 0.803944126167107e15 / 0.99709864531200e14 -0.19547569411550791e17 / 0.15704303663664000e17; 0.138091642084013e15 / 0.59351109840000e14 -0.886174803100459e15 / 0.99709864531200e14 0.7204491902193671e16 / 0.623186653320000e15 0.348854811893087e15 / 0.249274661328000e15 -0.5693689108983593e16 / 0.249274661328000e15 0.73965842628398389e17 / 0.2492746613280000e16 -0.2184472662036043e16 / 0.124637330664000e15 0.46667e5 / 0.10000e5; -0.3775041725375197e16 / 0.4486943903904000e16 0.4333e4 / 0.1000e4 -0.24847093554379e14 / 0.3115933266600e13 0.895954627955053e15 / 0.224347195195200e15 0.803944126167107e15 / 0.99709864531200e14 -0.2184472662036043e16 / 0.124637330664000e15 0.37593640125444199e17 / 0.2243471951952000e16 -0.37e2 / 0.4e1; 0.9907210230881393e16 / 0.61072292025360000e17 -0.13800578064893047e17 / 0.15704303663664000e17 0.943854037768721e15 / 0.545288321655000e15 -0.184881685054543e15 / 0.166183107552000e15 -0.19547569411550791e17 / 0.15704303663664000e17 0.46667e5 / 0.10000e5 -0.37e2 / 0.4e1 0.12766926490502478779e20 / 0.1099301256456480000e19;]; + + M4(1:8,1:8)=M4_U; + + M4(m-7:m,m-7:m)=flipud( fliplr( M4_U ) ); + M4=M4/h^3; + + S3_U=[-0.5e1 / 0.2e1 9 -12 7 -0.3e1 / 0.2e1;]/h^3; + S3_1=zeros(1,m); + S3_1(1:5)=S3_U; + S3_m=zeros(1,m); + S3_m(m-4:m)=fliplr(-S3_U); + + D4=HI*(M4-e_1*S3_1+e_m*S3_m + S_1'*S2_1-S_m'*S2_m); + + S_1 = S_1'; + S_m = S_m'; + S2_1 = S2_1'; + S2_m = S2_m'; + S3_1 = S3_1'; + S3_m = S3_m'; + + % L=h*(m-1); + + % x1=linspace(0,L,m)'; + % x2=x1.^2/fac(2); + % x3=x1.^3/fac(3); + % x4=x1.^4/fac(4); + % x5=x1.^5/fac(5); + + % x0=x1.^0/fac(1); + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/ordinary10.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,50 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = ordinary10(m,h) + + H_U = [0.5261271563e10 / 0.18289152000e11 0 0 0 0 0 0 0 0 0 0; 0 0.2881040311e10 / 0.1828915200e10 0 0 0 0 0 0 0 0 0; 0 0 0.52175551e8 / 0.406425600e9 0 0 0 0 0 0 0 0; 0 0 0 0.11662993e8 / 0.6096384e7 0 0 0 0 0 0 0; 0 0 0 0 0.50124587e8 / 0.87091200e8 0 0 0 0 0 0; 0 0 0 0 0 0.50124587e8 / 0.72576000e8 0 0 0 0 0; 0 0 0 0 0 0 0.148333439e9 / 0.87091200e8 0 0 0 0; 0 0 0 0 0 0 0 0.63867949e8 / 0.152409600e9 0 0 0; 0 0 0 0 0 0 0 0 0.20608675e8 / 0.16257024e8 0 0; 0 0 0 0 0 0 0 0 0 0.1704508063e10 / 0.1828915200e10 0; 0 0 0 0 0 0 0 0 0 0 0.18425967263e11 / 0.18289152000e11;]; + + H=eye(m); + H(1:11,1:11)=H_U; + H(m-10:m,m-10:m)=flipud( fliplr(H_U(1:11,1:11) ) ); + H=H*h; + HI=inv(H); + + Q=-(-1/1260*diag(ones(m-5,1),5)+5/504*diag(ones(m-4,1),4)-5/84*diag(ones(m-3,1),3)+5/21*diag(ones(m-2,1),2)-5/6*diag(ones(m-1,1),1)+5/6*diag(ones(m-1,1),-1)-5/21*diag(ones(m-2,1),-2)+5/84*diag(ones(m-3,1),-3)-5/504*diag(ones(m-4,1),-4)+1/1260*diag(ones(m-5,1),-5)); + + Q_U = [-0.1e1 / 0.2e1 0.2300876759589119e16 / 0.3395198177280000e16 -0.99808615498093e14 / 0.2263465451520000e16 -0.34957747037683e14 / 0.212199886080000e15 -0.709586095717e12 / 0.13473008640000e14 0.325330433051e12 / 0.6218311680000e13 0.27953548723573e14 / 0.485028311040000e15 0.2690678501e10 / 0.412439040000e12 -0.2397491025029e13 / 0.70733295360000e14 -0.9959492094287e13 / 0.1131732725760000e16 0.5242772857661e13 / 0.522338181120000e15; -0.2300876759589119e16 / 0.3395198177280000e16 0 0.3103439505511e13 / 0.16643128320000e14 0.2700334568377e13 / 0.5052378240000e13 0.50587599589937e14 / 0.242514155520000e15 -0.5570893587157e13 / 0.40419025920000e14 -0.1496329934863e13 / 0.8083805184000e13 -0.322512443237e12 / 0.12482346240000e14 0.2275340833763e13 / 0.23096586240000e14 0.22922115021893e14 / 0.848799544320000e15 -0.143e3 / 0.5000e4; 0.99808615498093e14 / 0.2263465451520000e16 -0.3103439505511e13 / 0.16643128320000e14 0 0.15053664233879e14 / 0.40419025920000e14 -0.9306441440587e13 / 0.32335220736000e14 -0.945459729233e12 / 0.13473008640000e14 0.956829267413e12 / 0.5774146560000e13 0.446866085903e12 / 0.56586636288000e14 -0.41109372242993e14 / 0.754488483840000e15 0.1e1 / 0.500e3 0.17e2 / 0.2500e4; 0.34957747037683e14 / 0.212199886080000e15 -0.2700334568377e13 / 0.5052378240000e13 -0.15053664233879e14 / 0.40419025920000e14 0 0.3899174751943e13 / 0.10104756480000e14 0.4691717443831e13 / 0.10104756480000e14 -0.58571891887e11 / 0.396264960000e12 0.100791910589e12 / 0.1040195520000e13 -0.425149181e9 / 0.29719872000e11 -0.2376515922259e13 / 0.30314269440000e14 0.36894656431e11 / 0.1036385280000e13; 0.709586095717e12 / 0.13473008640000e14 -0.50587599589937e14 / 0.242514155520000e15 0.9306441440587e13 / 0.32335220736000e14 -0.3899174751943e13 / 0.10104756480000e14 0 -0.4552305973e10 / 0.444165120000e12 0.4984940784247e13 / 0.11548293120000e14 -0.19410791e8 / 0.146764800e9 -0.2912773695913e13 / 0.40419025920000e14 0.127067639161e12 / 0.3233522073600e13 -0.89277540287e11 / 0.37309870080000e14; -0.325330433051e12 / 0.6218311680000e13 0.5570893587157e13 / 0.40419025920000e14 0.945459729233e12 / 0.13473008640000e14 -0.4691717443831e13 / 0.10104756480000e14 0.4552305973e10 / 0.444165120000e12 0 0.31722122083e11 / 0.84913920000e11 -0.887187251021e12 / 0.10104756480000e14 -0.1661755478749e13 / 0.26946017280000e14 0.1505713246249e13 / 0.13473008640000e14 -0.38859042469e11 / 0.1036385280000e13; -0.27953548723573e14 / 0.485028311040000e15 0.1496329934863e13 / 0.8083805184000e13 -0.956829267413e12 / 0.5774146560000e13 0.58571891887e11 / 0.396264960000e12 -0.4984940784247e13 / 0.11548293120000e14 -0.31722122083e11 / 0.84913920000e11 0 0.9357094407023e13 / 0.20209512960000e14 0.52602356173249e14 / 0.161676103680000e15 -0.1435252677707e13 / 0.17322439680000e14 -0.33048158431e11 / 0.3109155840000e13; -0.2690678501e10 / 0.412439040000e12 0.322512443237e12 / 0.12482346240000e14 -0.446866085903e12 / 0.56586636288000e14 -0.100791910589e12 / 0.1040195520000e13 0.19410791e8 / 0.146764800e9 0.887187251021e12 / 0.10104756480000e14 -0.9357094407023e13 / 0.20209512960000e14 0 0.70089734285659e14 / 0.141466590720000e15 -0.105938137621e12 / 0.471555302400e12 0.4358988450443e13 / 0.65292272640000e14; 0.2397491025029e13 / 0.70733295360000e14 -0.2275340833763e13 / 0.23096586240000e14 0.41109372242993e14 / 0.754488483840000e15 0.425149181e9 / 0.29719872000e11 0.2912773695913e13 / 0.40419025920000e14 0.1661755478749e13 / 0.26946017280000e14 -0.52602356173249e14 / 0.161676103680000e15 -0.70089734285659e14 / 0.141466590720000e15 0 0.314274398580227e15 / 0.377244241920000e15 -0.97822819709e11 / 0.487710720000e12; 0.9959492094287e13 / 0.1131732725760000e16 -0.22922115021893e14 / 0.848799544320000e15 -0.1e1 / 0.500e3 0.2376515922259e13 / 0.30314269440000e14 -0.127067639161e12 / 0.3233522073600e13 -0.1505713246249e13 / 0.13473008640000e14 0.1435252677707e13 / 0.17322439680000e14 0.105938137621e12 / 0.471555302400e12 -0.314274398580227e15 / 0.377244241920000e15 0 0.7519148725913e13 / 0.9327467520000e13; -0.5242772857661e13 / 0.522338181120000e15 0.143e3 / 0.5000e4 -0.17e2 / 0.2500e4 -0.36894656431e11 / 0.1036385280000e13 0.89277540287e11 / 0.37309870080000e14 0.38859042469e11 / 0.1036385280000e13 0.33048158431e11 / 0.3109155840000e13 -0.4358988450443e13 / 0.65292272640000e14 0.97822819709e11 / 0.487710720000e12 -0.7519148725913e13 / 0.9327467520000e13 0;]; + + Q(1:11,1:11)=Q_U; + Q(m-10:m,m-10:m)=flipud( fliplr(-Q_U ) ); + + D1=HI*Q; + + + s18=-1.000000000000000; s19=0.195000000000000; % alpha 0.0605 + s18= -0.475000000000000; s19=0.110000000000000; % alpha 0.0350 + %s18=0;s19=0; + DS=zeros(m,m); + DS(1,1:9)=[0.49e2 / 0.20e2 - s18 - (7 * s19) -0.6e1 + 0.7e1 * s18 + (48 * s19) 0.15e2 / 0.2e1 - 0.21e2 * s18 - (140 * s19) -0.20e2 / 0.3e1 + 0.35e2 * s18 + (224 * s19) 0.15e2 / 0.4e1 - 0.35e2 * s18 - (210 * s19) -0.6e1 / 0.5e1 + 0.21e2 * s18 + (112 * s19) 0.1e1 / 0.6e1 - 0.7e1 * s18 - (28 * s19) s18 s19;]; + DS(m,m-8:m)=fliplr(DS(1,1:9)); + DS=DS/h; + + M_U = [0.12056593789671863908e1 -0.13378814169347239658e1 0.36847309286546532061e-2 0.15698288365600946515e0 -0.37472461482539197952e-2 -0.62491712449361657064e-2 -0.29164045872729581661e-1 0.54848184117832929161e-3 0.13613461413384884448e-1 -0.25059220258337808220e-2 -0.94113457993630916498e-3; -0.13378814169347239658e1 0.21749807117105597139e1 -0.12369059547124894597e0 -0.83712574037924152603e0 0.50065127254670973258e-1 0.81045853127317536361e-2 0.97405846039248226536e-1 -0.68942461520402214720e-3 -0.41326971493379188475e-1 0.75778529605774119402e-2 0.25800256160095691057e-2; 0.36847309286546532061e-2 -0.12369059547124894597e0 0.18361596652499065332e0 0.48289690013342693109e-1 -0.19719621435164680412e0 0.11406859029505842791e0 -0.29646295985488126964e-1 -0.16038463172861201306e-2 0.32879841528337653050e-2 -0.93242311589807387463e-3 0.12241332668787820533e-3; 0.15698288365600946515e0 -0.83712574037924152603e0 0.48289690013342693109e-1 0.12886524606662484673e1 -0.14403037739488789185e0 -0.44846291607489015475e0 -0.10598334599408054277e0 -0.15873275740355918053e-1 0.73988493386459608166e-1 -0.12508848749152899785e-1 -0.39290233894513005339e-2; -0.37472461482539197952e-2 0.50065127254670973258e-1 -0.19719621435164680412e0 -0.14403037739488789185e0 0.51482665719685186210e0 0.51199577887125103015e-1 -0.36233561810883077365e0 0.91356850268746392169e-1 0.24195916108052419451e-2 -0.18564214413731389338e-2 -0.70192677320704413827e-3; -0.62491712449361657064e-2 0.81045853127317536361e-2 0.11406859029505842791e0 -0.44846291607489015475e0 0.51199577887125103015e-1 0.68636003380365860083e0 -0.28358848290867614908e0 -0.13836006478253396528e0 0.76158070663111995297e-2 0.11447010307180005164e-1 -0.21349696610286552676e-2; -0.29164045872729581661e-1 0.97405846039248226536e-1 -0.29646295985488126964e-1 -0.10598334599408054277e0 -0.36233561810883077365e0 -0.28358848290867614908e0 0.15216081480839085990e1 -0.42653865162216293237e0 -0.42047484981879143123e0 0.19813359263872926304e-1 0.19221397241190103344e-1; 0.54848184117832929161e-3 -0.68942461520402214720e-3 -0.16038463172861201306e-2 -0.15873275740355918053e-1 0.91356850268746392169e-1 -0.13836006478253396528e0 -0.42653865162216293237e0 0.10656733504627815335e1 -0.66921872668484232217e0 0.12022033144141336599e0 -0.30157881394591483631e-1; 0.13613461413384884448e-1 -0.41326971493379188475e-1 0.32879841528337653050e-2 0.73988493386459608166e-1 0.24195916108052419451e-2 0.76158070663111995297e-2 -0.42047484981879143123e0 -0.66921872668484232217e0 0.24064247712949611684e1 -0.15150200315922263367e1 0.17373015320416595052e0; -0.25059220258337808220e-2 0.75778529605774119402e-2 -0.93242311589807387463e-3 -0.12508848749152899785e-1 -0.18564214413731389338e-2 0.11447010307180005164e-1 0.19813359263872926304e-1 0.12022033144141336599e0 -0.15150200315922263367e1 0.27682502485427255096e1 -0.15975407111468405444e1; -0.94113457993630916498e-3 0.25800256160095691057e-2 0.12241332668787820533e-3 -0.39290233894513005339e-2 -0.70192677320704413827e-3 -0.21349696610286552676e-2 0.19221397241190103344e-1 -0.30157881394591483631e-1 0.17373015320416595052e0 -0.15975407111468405444e1 0.29033627686681129471e1;]; + + + M=-(1/3150)*diag(ones(m-5,1),5)+(5/1008)*diag(ones(m-4,1),4)-(5/126)*diag(ones(m-3,1),3)+(5/21)*diag(ones(m-2,1),2)-(5/3)*diag(ones(m-1,1),1)... + -(1/3150)*diag(ones(m-5,1),-5)+(5/1008)*diag(ones(m-4,1),-4)-(5/126)*diag(ones(m-3,1),-3)+(5/21)*diag(ones(m-2,1),-2)-(5/3)*diag(ones(m-1,1),-1)... + +(5269/1800)*diag(ones(m,1),0); + + M(1:11,1:11)=M_U; + M(m-10:m,m-10:m)=flipud( fliplr(M_U(1:11,1:11) ) ); + + D2=HI*(-M/h+DS); + + e_1 = zeros(m,1); + e_1(1)= 1; + e_m = zeros(m,1); + e_m(end)= 1; + S_1 = -DS(1,:)'; + S_m = DS(end,:)'; + + Q = H*D1-(-e_1*e_1' + e_m*e_m'); + M = -(H*D2-(-e_1*S_1' + e_m*S_m')); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/ordinary2.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,31 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = ordinary2(m,h) + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + H=(eye(m,m));H(1,1)=0.5;H(m,m)=0.5; + H=h*H; + HI=inv(H); + + D1=((.5*diag(ones(m-1,1),1)-.5*diag(ones(m-1,1),-1))); + 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; + + Q=H*D1 + 1/2*e_1*e_1' - 1/2*e_m*e_m'; + + D2=((diag(ones(m-1,1),1)+diag(ones(m-1,1),-1)-2*diag(ones(m,1),0))); + 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=zeros(1,m); + S_1(1:3)=S_U; + S_m=zeros(1,m); + S_m(m-2:m)=fliplr(-S_U); + + + 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/ordinary4.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,43 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = ordinary4(m,h) + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + e=ones(m,1); + H=spdiags(e,0,m,m); + %H=diag(ones(m,1),0); + + 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); + + + 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(1:4,1:4)=Q_U; + Q(m-3:m,m-3:m)=rot90( -Q_U(1:4,1:4) ,2 ); + + D1=HI*(Q-1/2*e_1*e_1'+1/2*e_m*e_m') ; + + M=-spdiags([-e 16*e -30*e 16*e -e], -2:2, m, m)/12; + + %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(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(zeros(1,m)); + S_1(1:4)=S_U; + S_m=sparse(zeros(1,m)); + S_m(m-3:m)=fliplr(-S_U); + + + D2=HI*(-M-e_1*S_1+e_m*S_m); + S_1 = S_1'; + S_m = S_m'; +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/ordinary6.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,53 @@ +function [H, HI, D1, D2, e_1, e_m, M,Q S_1, S_m] = ordinary6(m,h) + e_1=zeros(m,1);e_1(1)=1; + e_m=zeros(m,1);e_m(m)=1; + + H=diag(ones(m,1),0); + 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); + + H=H*h; + HI=inv(H); + + + % D1 har en fri parameter x1. + % Ett optimerat varde ger x1=0.70127127127127 = 331/472 + 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; + + Q=H*D1 + 1/2*e_1*e_1' - 1/2*e_m*e_m'; + + D2=(2*diag(ones(m-3,1),3)-27*diag(ones(m-2,1),2)+270*diag(ones(m-1,1),1)+270*diag(ones(m-1,1),-1)-27*diag(ones(m-2,1),-2)+2*diag(ones(m-3,1),-3)-490*diag(ones(m,1),0))/180; + + D2(1:6,1:9)=[114170/40947, -438107/54596, 336409/40947, -276997/81894, 3747/13649, 21035/163788, 0, 0, 0;6173/5860, -2066/879, 3283/1758, -303/293, 2111/3516, -601/4395, 0, 0, 0;-52391/81330, 134603/32532, -21982/2711, 112915/16266, -46969/16266, 30409/54220, 0, 0, 0;68603/321540, -12423/10718, 112915/32154, -75934/16077, 53369/21436, -54899/160770, 48/5359, 0, 0;-7053/39385, 86551/94524, -46969/23631, 53369/15754, -87904/23631, 820271/472620, -1296/7877, 96/7877, 0;21035/525612, -24641/131403, 30409/87602, -54899/131403, 820271/525612, -117600/43801, 64800/43801, -6480/43801, 480/43801]; + D2(m-5:m,m-8:m)=rot90( D2(1:6,1:9) ,2 ); + + D2=D2/h^2; + + S_U=[-25/12, 4, -3, 4/3, -1/4]/h; + S_1=zeros(1,m); + S_1(1:5)=S_U; + S_m=zeros(1,m); + S_m(m-4:m)=fliplr(-S_U); + + + 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/ordinary8.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,109 @@ +function [H, HI, D1, D2, e_1, e_m, M, Q, S_1, S_m] = ordinary8(m,h) + H=diag(ones(m,1),0); + H(1:8,1:8)=diag([1498139/5080320, 1107307/725760, 20761/80640, 1304999/725760, 299527/725760, 103097/80640, 670091/725760, 5127739/5080320]); + H(m-7:m,m-7:m)=fliplr(flipud(diag([1498139/5080320, 1107307/725760, 20761/80640, 1304999/725760, 299527/725760, 103097/80640, 670091/725760, 5127739/5080320]))); + + D1=-(1/280*diag(ones(m-4,1),4)-4/105*diag(ones(m-3,1),3)+1/5*diag(ones(m-2,1),2)-4/5*diag(ones(m-1,1),1)+4/5*diag(ones(m-1,1),-1)-1/5*diag(ones(m-2,1),-2)+4/105*diag(ones(m-3,1),-3)-1/280*diag(ones(m-4,1),-4)); + + + + + %r68 = -1022551/30481920; + %r78 = 6445687/8709120; + %r67 = 1714837/4354560; + + %r67=0.58; + %r68=-0.08; + %r78=0.75; + + %r67=0.65; + %r68=-0.1; + %r78=0.75; + + %r67=0.9250; + %r68=-0.2; + %r78=0.775; + + %r67=0.65; + %r68=-0.105; + %r78=0.755; + + %%r67=0.649; + %%r68=-0.104; + %%r78=0.755; + + %r67=-0.48; + %r68=0.3; + %r78=0.67; + + %r67=0.5600; + %r68=-0.0733; + %r78=0.7500;/scr0/home/ken/VERY_FINE + + % min med 1/10 f?r D6 + %r67=0.62; + %r68=-0.1040; + %r78=0.7640; + + % Den nya optimerade, for att fungera i NS-dissipation + + r67=0.69789473684211; + r68=-0.12052631578947; + r78=0.75868421052632; + + + D1(1:8,1:12)=[-2540160/1498139, -142642467/5992556+50803200/1498139*r78+5080320/1498139*r67+25401600/1498139*r68, 705710031/5992556-228614400/1498139*r78-25401600/1498139*r67-121927680/1498139*r68, -3577778591/17977668+381024000/1498139*r78+50803200/1498139*r67+228614400/1498139*r68, 203718909/1498139-254016000/1498139*r78-50803200/1498139*r67-203212800/1498139*r68, -32111205/5992556+25401600/1498139*r67+76204800/1498139*r68, -652789417/17977668+76204800/1498139*r78-5080320/1498139*r67, 74517981/5992556-25401600/1498139*r78-5080320/1498139*r68, 0, 0, 0, 0;142642467/31004596-7257600/1107307*r78-725760/1107307*r67-3628800/1107307*r68, 0, -141502371/2214614+91445760/1107307*r78+10886400/1107307*r67+50803200/1107307*r68, 159673719/1107307-203212800/1107307*r78-29030400/1107307*r67-127008000/1107307*r68, -1477714693/13287684+152409600/1107307*r78+32659200/1107307*r67+127008000/1107307*r68, 11652351/2214614-17418240/1107307*r67-50803200/1107307*r68, 36069450/1107307-50803200/1107307*r78+3628800/1107307*r67, -536324953/46506894+17418240/1107307*r78+3628800/1107307*r68, 0, 0, 0, 0;-18095129/134148+3628800/20761*r78+403200/20761*r67+1935360/20761*r68, 47167457/124566-10160640/20761*r78-1209600/20761*r67-5644800/20761*r68, 0, -120219461/124566+25401600/20761*r78+4032000/20761*r67+16934400/20761*r68, 249289259/249132-25401600/20761*r78-6048000/20761*r67-22579200/20761*r68, -2611503/41522+3628800/20761*r67+10160640/20761*r68, -7149666/20761+10160640/20761*r78-806400/20761*r67, 37199165/290654-3628800/20761*r78-806400/20761*r68, 0, 0, 0, 0;3577778591/109619916-54432000/1304999*r78-7257600/1304999*r67-32659200/1304999*r68, -159673719/1304999+203212800/1304999*r78+29030400/1304999*r67+127008000/1304999*r68, 360658383/2609998-228614400/1304999*r78-36288000/1304999*r67-152409600/1304999*r68, 0, -424854441/5219996+127008000/1304999*r78+36288000/1304999*r67+127008000/1304999*r68, 22885113/2609998-29030400/1304999*r67-76204800/1304999*r68, 158096578/3914997-76204800/1304999*r78+7257600/1304999*r67, -296462325/18269986+29030400/1304999*r78+7257600/1304999*r68, 0, 0, 0, 0;-203718909/2096689+36288000/299527*r78+7257600/299527*r67+29030400/299527*r68, 1477714693/3594324-152409600/299527*r78-32659200/299527*r67-127008000/299527*r68, -747867777/1198108+228614400/299527*r78+54432000/299527*r67+203212800/299527*r68, 424854441/1198108-127008000/299527*r78-36288000/299527*r67-127008000/299527*r68, 0, -17380335/1198108+10886400/299527*r67+25401600/299527*r68, -67080435/1198108+25401600/299527*r78-3628800/299527*r67, 657798011/25160268-10886400/299527*r78-3628800/299527*r68, -2592/299527, 0, 0, 0;1529105/1237164-403200/103097*r67-1209600/103097*r68, -3884117/618582+1935360/103097*r67+5644800/103097*r68, 2611503/206194-3628800/103097*r67-10160640/103097*r68, -7628371/618582+3225600/103097*r67+8467200/103097*r68, 5793445/1237164-1209600/103097*r67-2822400/103097*r68, 0, 80640/103097*r67, 80640/103097*r68, 3072/103097, -288/103097, 0, 0;93255631/8041092-10886400/670091*r78+725760/670091*r67, -36069450/670091+50803200/670091*r78-3628800/670091*r67, 64346994/670091-91445760/670091*r78+7257600/670091*r67, -158096578/2010273+76204800/670091*r78-7257600/670091*r67, 67080435/2680364-25401600/670091*r78+3628800/670091*r67, -725760/670091*r67, 0, 725760/670091*r78, -145152/670091, 27648/670091, -2592/670091, 0;-3921999/1079524+25401600/5127739*r78+5080320/5127739*r68, 536324953/30766434-121927680/5127739*r78-25401600/5127739*r68, -334792485/10255478+228614400/5127739*r78+50803200/5127739*r68, 296462325/10255478-203212800/5127739*r78-50803200/5127739*r68, -657798011/61532868+76204800/5127739*r78+25401600/5127739*r68, -5080320/5127739*r68, -5080320/5127739*r78, 0, 4064256/5127739, -1016064/5127739, 193536/5127739, -18144/5127739]; + + + D1(m-7:m,m-11:m)=flipud( fliplr(-D1(1:8,1:12))); + + + D1=D1/h; + + %DD=-(1/280*diag(ones(m-4,1),4)-4/105*diag(ones(m-3,1),3)+1/5*diag(ones(m-2,1),2)-4/5*diag(ones(m-1,1),1)+4/5*diag(ones(m-1,1),-1)-1/5*diag(ones(m-2,1),-2)+4/105*diag(ones(m-3,1),-3)-1/280*diag(ones(m-4,1),-4)); + %DD(1:4,1:9)=1/280*[-761,2240,-3920,15680/3,-4900,3136,-3920/3,320,-35, + % -35,-446,980,-980,2450/3,-490,196,-140/3,5, + % 5,-80,-266,560,-350,560/3,-70,16,-5/3, + % -5/3,20,-140,-126,350,-140,140/3,-10,1]; + D(m-7:m,m-11:m)=flipud( fliplr(-D1(1:8,1:12))); + + D2=(-1/560*diag(ones(m-4,1),4)+8/315*diag(ones(m-3,1),3)-1/5*diag(ones(m-2,1),2)+8/5*diag(ones(m-1,1),1)+8/5*diag(ones(m-1,1),-1)-1/5*diag(ones(m-2,1),-2)+8/315*diag(ones(m-3,1),-3)-1/560*diag(ones(m-4,1),-4)-205/72*diag(ones(m,1),0)); + + D2(1:8,1:12)=[4870382994799/1358976868290, -893640087518/75498714905,926594825119/60398971924, -1315109406200/135897686829,39126983272/15099742981, 12344491342/75498714905, -451560522577/2717953736580, 0, 0, 0, 0, 0;333806012194/390619153855, -154646272029/111605472530, 1168338040/33481641759, 82699112501/133926567036, -171562838/11160547253, -28244698346/167408208795, 11904122576/167408208795, -2598164715/312495323084, 0, 0, 0, 0;7838984095/52731029988, 1168338040/5649753213, -88747895/144865467, 423587231/627750357, -43205598281/22599012852, 4876378562/1883251071, -5124426509/3766502142, 10496900965/39548272491, 0, 0, 0, 0;-94978241528/828644350023, 82699112501/157837019052, 1270761693/13153084921, -167389605005/118377764289, 48242560214/39459254763, -31673996013/52612339684, 43556319241/118377764289, -44430275135/552429566682, 0, 0, 0, 0;1455067816/21132528431, -171562838/3018932633, -43205598281/36227191596, 48242560214/9056797899, -52276055645/6037865266, 57521587238/9056797899, -80321706377/36227191596, 8078087158/21132528431, -1296/299527, 0, 0, 0;10881504334/327321118845, -28244698346/140280479505, 4876378562/9352031967, -10557998671/12469375956, 57521587238/28056095901, -278531401019/93520319670, 73790130002/46760159835, -137529995233/785570685228, 2048/103097, -144/103097, 0, 0;-135555328849/8509847458140, 11904122576/101307707835, -5124426509/13507694378, 43556319241/60784624701, -80321706377/81046166268, 73790130002/33769235945, -950494905688/303923123505, 239073018673/141830790969, -145152/670091, 18432/670091, -1296/670091, 0;0, -2598164715/206729925524, 10496900965/155047444143, -44430275135/310094888286, 425162482/2720130599, -137529995233/620189776572, 239073018673/155047444143, -144648000000/51682481381, 8128512/5127739, -1016064/5127739, 129024/5127739, -9072/5127739]; + + + + + D2(m-7:m,m-11:m)=flipud( fliplr(D2(1:8,1:12) ) ); + + D2=D2/h^2; + + DS=zeros(m,m); + DS(1,1:7)=-[-4723/2100, 839/175, -157/35, 278/105, -103/140, -1/175, 6/175]; + + DS(m,m-6:m)=fliplr(-[-4723/2100, 839/175, -157/35, 278/105, -103/140, -1/175, 6/175]); + DS=DS/h; + + H=h*H; + HI=inv(H); + + %r1=D1*u-u_x;sqrt(r1'*r1)/m + %r2=D2*u-u_xx;sqrt(r2'*r2)/m + + %te=eig(D1); + %tm=max(abs(te)); + %plot(real(te),imag(te),'*'); + %grid; + %xlabel('Real part'); + %ylabel('Imaginary part'); + %title('Spectrum, minimal spectral radius'); + e_1 = zeros(m,1); + e_1(1)= 1; + e_m = zeros(m,1); + e_m(end)= 1; + S_1 = -DS(1,:)'; + S_m = DS(end,:)'; + + Q = H*D1-(-e_1*e_1' + e_m*e_m'); + M = -(H*D2-(-e_1*S_1' + e_m*S_m')); +end \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/+sbp/variable4.m Mon Sep 28 08:47:28 2015 +0200 @@ -0,0 +1,115 @@ +function [H, HI, D1, D2, e_l, e_r, d_l, d_r] = variable4(m,h) + N = m; + + H = speye(N); + H(1,1) = 17/48; H(2,2) = 59/48; H(3,3) = 43/48; H(4,4) = 49/48; + H(N,N) = 17/48; H(N-1,N-1) = 59/48; H(N-2,N-2) = 43/48; H(N-3,N-3) = 49/48; + H = h*H; + + HI = inv(H); + + H = sparse(H); + HI = sparse(HI); + + + S = sparse(N,N); + S(1,1:4) = 1/h*[(-11/6);3;(-3/2);1/3]; + S(N,N-3:N) = 1/h*[(-1/3);3/2;(-3);11/6]; + S = sparse(S); + + e_l = sparse(m,1); + e_l(1) = 1; + + e_r = sparse(m,1); + e_r(end) = 1; + + d_l = S(1,:)'; + d_r = S(end,:)'; + + 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 ); + + D1=HI*(Q-1/2*e_l*e_l'+1/2*e_r*e_r') ; + + function D2 = D2_fun(c) + M = 78+(N-12)*5; + %h = 1/(N-1); + + + U = [48/(17)*(0.12e2 / 0.17e2 * c(1) + 0.59e2 / 0.192e3 * c(2) + 0.27010400129e11 / 0.345067064608e12 * c(3) + 0.69462376031e11 / 0.2070402387648e13 * c(4)) 48/(17)*(-0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4)) 48/(17)*(0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3)) 48/(17)*(0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4)) 48/(17)*(0.49579087e8 / 0.10149031312e11 * c(3) - 0.49579087e8 / 0.10149031312e11 * c(4)) 48/(17)*(-c(4) / 0.784e3 + c(3) / 0.784e3);... + 48/(59)*(-0.59e2 / 0.68e2 * c(1) - 0.6025413881e10 / 0.21126554976e11 * c(3) - 0.537416663e9 / 0.7042184992e10 * c(4)) 48/(59)*(0.3481e4 / 0.3264e4 * c(1) + 0.9258282831623875e16 / 0.7669235228057664e16 * c(3) + 0.236024329996203e15 / 0.1278205871342944e16 * c(4)) 48/(59)*(-0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4)) 48/(59)*(-0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4)) 48/(59)*(-0.1328188692663e13 / 0.37594290333616e14 * c(3) + 0.1328188692663e13 / 0.37594290333616e14 * c(4)) 48/(59)*(-0.8673e4 / 0.2904112e7 * c(3) + 0.8673e4 / 0.2904112e7 * c(4));... + 48/(43)*(0.2e1 / 0.17e2 * c(1) - 0.59e2 / 0.192e3 * c(2) + 0.213318005e9 / 0.16049630912e11 * c(4) + 0.2083938599e10 / 0.8024815456e10 * c(3)) 48/(43)*(-0.59e2 / 0.408e3 * c(1) - 0.29294615794607e14 / 0.29725717938208e14 * c(3) - 0.2944673881023e13 / 0.29725717938208e14 * c(4)) 48/(43)*(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)) 48/(43)*(c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4)) 48/(43)*(-0.10532412077335e14 / 0.42840005263888e14 * c(4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(3) + 0.564461e6 / 0.4461432e7 * c(5)) 48/(43)*(-0.960119e6 / 0.1280713392e10 * c(4) - 0.3391e4 / 0.6692148e7 * c(5) + 0.33235054191e11 / 0.26452850508784e14 * c(3));... + 48/(49)*(0.3e1 / 0.68e2 * c(1) - 0.1244724001e10 / 0.21126554976e11 * c(3) + 0.752806667e9 / 0.21126554976e11 * c(4)) 48/(49)*(-0.59e2 / 0.1088e4 * c(1) + 0.260297319232891e15 / 0.2556411742685888e16 * c(3) - 0.60834186813841e14 / 0.1278205871342944e16 * c(4)) 48/(49)*(c(1) / 0.136e3 - 0.125059e6 / 0.743572e6 * c(5) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(3) - 0.17220493277981e14 / 0.89177153814624e14 * c(4)) 48/(49)*(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)) 48/(49)*(-0.4959271814984644613e19 / 0.20965546238960637264e20 * c(3) - c(6) / 0.6e1 - 0.15998714909649e14 / 0.37594290333616e14 * c(4) - 0.375177e6 / 0.743572e6 * c(5)) 48/(49)*(-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)]; + + + L = [c(N-7) / 0.24e2 + 0.5e1 / 0.6e1 * c(N-6) + 0.5580181e7 / 0.6692148e7 * c(N-4) + 0.4887707739997e13 / 0.119037827289528e15 * c(N-3) + 0.3e1 / 0.4e1 * c(N-5) + 0.660204843e9 / 0.13226425254392e14 * c(N-2) + 0.660204843e9 / 0.13226425254392e14 * c(N-1) -c(N-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(N-3) - c(N-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(N-4) - 0.13091810925e11 / 0.13226425254392e14 * c(N-2) - 0.13091810925e11 / 0.13226425254392e14 * c(N-1) -0.368395e6 / 0.2230716e7 * c(N-4) + c(N-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(N-3) + 0.752806667e9 / 0.539854092016e12 * c(N-2) + 0.752806667e9 / 0.539854092016e12 * c(N-1) -0.3391e4 / 0.6692148e7 * c(N-4) - 0.238797444493e12 / 0.119037827289528e15 * c(N-3) + 0.33235054191e11 / 0.26452850508784e14 * c(N-2) + 0.33235054191e11 / 0.26452850508784e14 * c(N-1) -0.8673e4 / 0.2904112e7 * c(N-2) - 0.8673e4 / 0.2904112e7 * c(N-1) + 0.8673e4 / 0.1452056e7 * c(N-3) -c(N-3) / 0.392e3 + c(N-2) / 0.784e3 + c(N-1) / 0.784e3;... + -c(N-6) / 0.6e1 - 0.1618585929605e13 / 0.9919818940794e13 * c(N-3) - c(N-5) / 0.2e1 - 0.1118749e7 / 0.2230716e7 * c(N-4) - 0.13091810925e11 / 0.13226425254392e14 * c(N-2) - 0.13091810925e11 / 0.13226425254392e14 * c(N-1) c(N-6) / 0.24e2 + 0.5e1 / 0.6e1 * c(N-5) + 0.3896014498639e13 / 0.4959909470397e13 * c(N-3) + 0.8386761355510099813e19 / 0.128413970713633903242e21 * c(N-2) + 0.280535e6 / 0.371786e6 * c(N-4) + 0.3360696339136261875e19 / 0.171218627618178537656e21 * c(N-1) -c(N-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(N-2) - 0.375177e6 / 0.743572e6 * c(N-4) - 0.13425842714e11 / 0.33740880751e11 * c(N-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(N-1) -0.365281640980e12 / 0.1653303156799e13 * c(N-3) + 0.564461e6 / 0.4461432e7 * c(N-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(N-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(N-1) -0.1328188692663e13 / 0.37594290333616e14 * c(N-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(N-1) - 0.8673e4 / 0.363014e6 * c(N-3) c(N-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(N-2) - 0.256702175e9 / 0.10149031312e11 * c(N-1);... + 48/(49)*(-0.368395e6 / 0.2230716e7 * c(N-4) + c(N-5) / 0.8e1 + 0.48866620889e11 / 0.404890569012e12 * c(N-3) + 0.752806667e9 / 0.539854092016e12 * c(N-2) + 0.752806667e9 / 0.539854092016e12 * c(N-1)) 48/(49)*(-c(N-5) / 0.6e1 - 0.4959271814984644613e19 / 0.20965546238960637264e20 * c(N-2) - 0.375177e6 / 0.743572e6 * c(N-4) - 0.13425842714e11 / 0.33740880751e11 * c(N-3) - 0.193247108773400725e18 / 0.6988515412986879088e19 * c(N-1)) 48/(49)*(c(N-5) / 0.24e2 + 0.1869103e7 / 0.2230716e7 * c(N-4) + 0.507284006600757858213e21 / 0.475219048083107777984e21 * c(N-2) + 0.3e1 / 0.1088e4 * c(N) + 0.31688435395e11 / 0.67481761502e11 * c(N-3) + 0.27769176016102795561e20 / 0.712828572124661666976e21 * c(N-1)) 48/(49)*(-0.125059e6 / 0.743572e6 * c(N-4) + c(N) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(N-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(N-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(N-1)) 48/(49)*(0.260297319232891e15 / 0.2556411742685888e16 * c(N-2) - 0.59e2 / 0.1088e4 * c(N) - 0.106641839640553e15 / 0.1278205871342944e16 * c(N-1) + 0.26019e5 / 0.726028e6 * c(N-3)) 48/(49)*(-0.1244724001e10 / 0.21126554976e11 * c(N-2) + 0.3e1 / 0.68e2 * c(N) + 0.752806667e9 / 0.21126554976e11 * c(N-1));... + 48/(43)*(-0.3391e4 / 0.6692148e7 * c(N-4) - 0.238797444493e12 / 0.119037827289528e15 * c(N-3) + 0.33235054191e11 / 0.26452850508784e14 * c(N-2) + 0.33235054191e11 / 0.26452850508784e14 * c(N-1)) 48/(43)*(-0.365281640980e12 / 0.1653303156799e13 * c(N-3) + 0.564461e6 / 0.4461432e7 * c(N-4) + 0.1613976761032884305e19 / 0.7963657098519931984e19 * c(N-2) - 0.198407225513315475e18 / 0.7963657098519931984e19 * c(N-1)) 48/(43)*(-0.125059e6 / 0.743572e6 * c(N-4) + c(N) / 0.136e3 - 0.23099342648e11 / 0.101222642253e12 * c(N-3) - 0.4836340090442187227e19 / 0.5525802884687299744e19 * c(N-2) + 0.193950157930938693e18 / 0.5525802884687299744e19 * c(N-1)) 48/(43)*(0.564461e6 / 0.13384296e8 * c(N-4) + 0.470299699916357e15 / 0.952302618316224e15 * c(N-3) + 0.550597048646198778781e21 / 0.1624586048098066124736e22 * c(N-1) + c(N) / 0.51e2 + 0.378288882302546512209e21 / 0.270764341349677687456e21 * c(N-2)) 48/(43)*(-0.59e2 / 0.408e3 * c(N) - 0.29294615794607e14 / 0.29725717938208e14 * c(N-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(N-1) - 0.8673e4 / 0.363014e6 * c(N-3)) 48/(43)*(-0.59e2 / 0.3136e4 * c(N-3) - 0.13249937023e11 / 0.48148892736e11 * c(N-1) + 0.2e1 / 0.17e2 * c(N) + 0.2083938599e10 / 0.8024815456e10 * c(N-2));... + 48/(59)*(-0.8673e4 / 0.2904112e7 * c(N-2) - 0.8673e4 / 0.2904112e7 * c(N-1) + 0.8673e4 / 0.1452056e7 * c(N-3)) 48/(59)*(-0.1328188692663e13 / 0.37594290333616e14 * c(N-2) + 0.2226377963775e13 / 0.37594290333616e14 * c(N-1) - 0.8673e4 / 0.363014e6 * c(N-3)) 48/(59)*(0.260297319232891e15 / 0.2556411742685888e16 * c(N-2) - 0.59e2 / 0.1088e4 * c(N) - 0.106641839640553e15 / 0.1278205871342944e16 * c(N-1) + 0.26019e5 / 0.726028e6 * c(N-3)) 48/(59)*(-0.59e2 / 0.408e3 * c(N) - 0.29294615794607e14 / 0.29725717938208e14 * c(N-2) - 0.2234477713167e13 / 0.29725717938208e14 * c(N-1) - 0.8673e4 / 0.363014e6 * c(N-3)) 48/(59)*(0.9258282831623875e16 / 0.7669235228057664e16 * c(N-2) + 0.3481e4 / 0.3264e4 * c(N) + 0.228389721191751e15 / 0.1278205871342944e16 * c(N-1) + 0.8673e4 / 0.1452056e7 * c(N-3)) 48/(59)*(-0.6025413881e10 / 0.21126554976e11 * c(N-2) - 0.59e2 / 0.68e2 * c(N) - 0.537416663e9 / 0.7042184992e10 * c(N-1));... + 48/(17)*(-c(N-3) / 0.392e3 + c(N-2) / 0.784e3 + c(N-1) / 0.784e3) 48/(17)*(c(N-3) / 0.49e2 + 0.49579087e8 / 0.10149031312e11 * c(N-2) - 0.256702175e9 / 0.10149031312e11 * c(N-1)) 48/(17)*(-0.1244724001e10 / 0.21126554976e11 * c(N-2) + 0.3e1 / 0.68e2 * c(N) + 0.752806667e9 / 0.21126554976e11 * c(N-1)) 48/(17)*(-0.59e2 / 0.3136e4 * c(N-3) - 0.13249937023e11 / 0.48148892736e11 * c(N-1) + 0.2e1 / 0.17e2 * c(N) + 0.2083938599e10 / 0.8024815456e10 * c(N-2)) 48/(17)*(-0.6025413881e10 / 0.21126554976e11 * c(N-2) - 0.59e2 / 0.68e2 * c(N) - 0.537416663e9 / 0.7042184992e10 * c(N-1)) 48/(17)*(0.3e1 / 0.3136e4 * c(N-3) + 0.27010400129e11 / 0.345067064608e12 * c(N-2) + 0.234566387291e12 / 0.690134129216e12 * c(N-1) + 0.12e2 / 0.17e2 * c(N))]; + + + + R = zeros(M,1); + R(1:24) = reshape(U(1:4,:)',24,1); + R(25:30) = U(5,:); + R(31) = -c(5+1) / 0.6e1 + c(5) / 0.8e1 + c(5+2) / 0.8e1; + R(32:37) = U(6,:); + R(38:39) = [-c(6-1) / 0.6e1 - c(6+2) / 0.6e1 - c(6) / 0.2e1 - c(6+1) / 0.2e1;... + -c(6+1) / 0.6e1 + c(6) / 0.8e1 + c(6+2) / 0.8e1]; + + R(M-38:M-37) = [-c(N-6) / 0.6e1 + c(N-7) / 0.8e1 + c(N-5) / 0.8e1;... + -c(N-7) / 0.6e1 - c(N-4) / 0.6e1 - c(N-6) / 0.2e1 - c(N-5) / 0.2e1]; + R(M-36:M-31) = L(1,:); + R(M-30) = -c(N-5) / 0.6e1 + c(N-6) / 0.8e1 + c(N-4) / 0.8e1; + R(M-29:M-24) = L(2,:); + R(M-23:M) = reshape(L(3:6,:)',24,1); + + for i=7:N-6 + R(40+(i-7)*5:44+(i-7)*5) = [-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 + + R = R/h/h; + D2 = -R; + D2(1:4) = -48/17/h/h*[c(1)*(-11/6);c(1)*3;c(1)*(-3/2);c(1)*1/3] + D2(1:4); + D2(M-3:M) = -48/17/h/h*[c(N)*1/3;c(N)*(-3/2);c(N)*3;c(N)*(-11/6)] + D2(M-3:M); + + + + BS = sparse(N,N); + BS(1,1:4) = -c(1)*1/h*[(-11/6);3;(-3/2);1/3]; + BS(N,N-3:N) = c(N)*1/h*[(-1/3);3/2;(-3);11/6]; + BS = sparse(BS); + + % %%Row and column indices%% + M = 78+(N-12)*5; + rows = [kron([1;2;3;4],ones(6,1));... + 5*ones(7,1);... + 6*ones(8,1);... + kron((7:N-6)',ones(5,1));... + (N-5)*ones(8,1);... + (N-4)*ones(7,1);... + kron([N-3;N-2;N-1;N],ones(6,1))]; + + cols = zeros(M,1); + cols(1:24) = kron(ones(4,1),[1;2;3;4;5;6]); + cols(25:39) = [(1:7)';(1:8)']; + cols(M-23:M) = kron(ones(4,1),[N-5;N-4;N-3;N-2;N-1;N]); + cols(M-38:M-24) = [(N-7:N)';(N-6:N)']; + for i=7:N-6 + cols(40+(i-7)*5:44+(i-7)*5) = [i-2;i-1;i;i+1;i+2]; + end + D2 = sparse(rows,cols,D2); + end + D2 = @D2_fun; +end \ No newline at end of file