Mercurial > repos > public > sbplib
view +sbp/+implementations/d4_compatible_2.m @ 261:6009f2712d13 operator_remake
Moved and renamned all implementations.
author | Martin Almquist <martin.almquist@it.uu.se> |
---|---|
date | Thu, 08 Sep 2016 15:35:45 +0200 |
parents | |
children | bfa130b7abf6 |
line wrap: on
line source
function [H, HI, D1, D4, e_1, e_m, M4, Q, S2_1, S2_m,... S3_1, S3_m, S_1, S_m] = d4_compatible_2(m,h) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% 4:de ordn. SBP Finita differens %%% %%% operatorer framtagna av Ken Mattsson %%% %%% %%% %%% 6 randpunkter, diagonal norm %%% %%% %%% %%% Datum: 2013-11-11 %%% %%% %%% %%% %%% %%% H (Normen) %%% %%% D1 (approx f?rsta derivatan) %%% %%% D2 (approx andra derivatan) %%% %%% D3 (approx tredje derivatan) %%% %%% D2 (approx fj?rde derivatan) %%% %%% %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % M?ste ange antal punkter (m) och stegl?ngd (h) % Notera att dessa opetratorer ?r framtagna f?r att anv?ndas n?r % vi har 3de och 4de derivator i v?r PDE % I annat fall anv?nd de "traditionella" som har noggrannare % randsplutningar f?r D1 och D2 % Vi b?rjar med normen. Notera att alla SBP operatorer delar samma norm, % vilket ?r n?dv?ndigt f?r stabilitet 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