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ParaDIAG_V1_Hybrid_for_ADE.m
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ParaDIAG_V1_Hybrid_for_ADE.m
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clc;
clear;
global Nt Nx nu tau theta;
dt_max=1e-2;
theta=1/2;
tau=1.15;
nu=1e-3;
T=2;
dx=1/64;
dt=1/256;
x0=(-1:dx:1)';
%x=x0(2:end-1);
x=x0;
U0=sin(2*pi*x);
W0=U0;
Nx=length(U0);
Nx0=length(x);
Ix0=speye(Nx0);
Ix=speye(Nx);
A1=spdiags([-ones(Nx0,1),2*ones(Nx0,1),-ones(Nx0,1)]/dx^2,[-1,0,1],Nx0,Nx0);
A1(1,Nx0)=-1/dx^2;
A1(Nx0,1)=A1(1,Nx0);
A2=spdiags([-ones(Nx0,1),0*ones(Nx0,1),ones(Nx0,1)]/(2*dx),[-1,0,1],Nx0,Nx0);
A2(1,Nx0)=A2(2,1);
A2(Nx0,1)=A2(1,2);
A=nu*A1+A2;
NNt=4:2:60;
NK=length(NNt);
Err_Diag=zeros(2,NK);
CondDiag=zeros(2,NK);
TT=zeros(1,NK);
TT=zeros(1,NK);
for jn=1:NK
Nt=NNt(jn);
It=speye(Nt);
dt0=dt_max/tau^(Nt-1);
dtn=dt0*tau.^(0:Nt-1);
t=zeros(1,Nt+1);
for n=2:Nt+1
t(n)=sum(dtn(1:n-1));
end
T=t(end);
TT(jn)=T;
U_ref=zeros(Nx0,Nt+1);
U_ref(:,1)=U0;
U_sbs=zeros(Nx0,Nt+1);
U_sbs(:,1)=U0;
for n=1:Nt
U_sbs(:,n+1)=((Ix+dtn(n)*theta*A)\(Ix-dtn(n)*(1-theta)*A))*U_sbs(:,n);
U_ref(:,n+1)=expm(-A*t(n+1))*U0;
end
%% U_ParaDIAG1: solution computed by the ParaDIAG-I algorithm proposed by Martin SISC-2019
B1=zeros(Nt,Nt);
for n=1:Nt
B1(n,n)=1/dtn(n);
end
for n=1:Nt-1
B1(n+1,n)=-1/dtn(n+1);
end
B2=spdiags([(1-theta)*ones(Nt,1),theta*ones(Nt,1)],[-1,0],Nt,Nt);
b=kron(B2\It,Ix)*[(Ix/dtn(1)-(1-theta)*A)*W0;zeros((Nt-1)*Nx,1)];
U_ParaDIAG1=zeros(Nx,Nt+1);
U_ParaDIAG1(:,1)=U0;
[V,D,invV]=getV(dtn);
sol_stepA=StepAC(invV,b,Nx);
sol_stepB=zeros(Nx*Nt,1);
for n=1:Nt
sol_stepB((n-1)*Nx+1:n*Nx,1)=(D(n)*Ix+A)\sol_stepA((n-1)*Nx+1:n*Nx,1);
end
sol_stepC=StepAC(V,sol_stepB,Nx);
U_ParaDIAG1(:,2:Nt+1)=reshape(sol_stepC,Nx,Nt);
%% U_ParaDIAG2: solution computed by the ParaDIAG-2 algorithm proposed new
dt=T/Nt;
[V,D0,invV,iter]=fasteigB(Nt,dt,1e-9);
D=diag(D0);
b=[U0/(2*dt);zeros((Nt-1)*Nx0,1)];
U_ParaDIAG2=zeros(Nx0,Nt+1);
U_ParaDIAG2(:,1)=U0;
stepA=StepAC(invV,b,Nx0);
stepB=zeros(Nx0*Nt,1);
U_refnew=zeros(Nx0,Nt+1);
U_refnew(:,1)=U0;
for n=1:Nt
stepB((n-1)*Nx0+1:n*Nx0,1)=(D(n,n)*Ix0+A)\stepA((n-1)*Nx0+1:n*Nx0,1);
U_refnew(:,n+1)=expm(-(n*dt)*A)*U0;
end
stepC=StepAC(V,stepB,Nx0);
U_ParaDIAG2(:,2:Nt+1)=reshape(stepC,Nx0,Nt);
Err_sbs(jn)=max(max(abs((U_ref-U_sbs))));
Err_Diag(1,jn)=max(max(abs((U_ref-U_ParaDIAG1(1:Nx0,:)))));
Err_Diag(2,jn)=max(max(abs((U_refnew-U_ParaDIAG2))));
figure(2);
semilogy(NNt(1:jn),Err_Diag(2,1:jn),'r-.s',NNt(1:jn),Err_Diag(1,1:jn),'b-.o',NNt(1:jn),Err_sbs(1:jn),'k',NNt(1:jn),TT(1:jn),'w-d','markersize',8,'linewidth',1);shg;
set(gca,'fontname','Times New Roman','fontsize',13);
xlabel('$N_t$','interpreter','latex','fontsize',18);
ylabel('global error','interpreter','latex','fontsize',18);
leg=legend('ParaDiag-I using $\Delta t$ (new)', 'ParaDiag-I using $\Delta t_n=\Delta t_{N_t}\tau^{n-N_t}$','Time-stepping TR using $\Delta t_n=\Delta t_{N_t}\tau^{n-N_t}$');
set(leg,'interpreter','latex','fontsize',15,'location','northwest');
%title(['$T=$',num2str(T)],'interpreter','latex','fontsize',20);
xlim([NNt(1)-1,NNt(end)+1]);
end
%mesh(t_ode45(1:20:length(t_ode45)),x,U_ode45(:,1:20:length(t_ode45)));shg
% nt=60;
% mesh(t(1:nt),x,U_ParaDIAG1(:,1:nt));shg
%
% set(gca,'fontname','Times New Roman','fontsize',12);
% xlabel('time: t','fontname','Times New Roman','fontsize',20);
% ylabel('space: x','fontname','Times New Roman','fontsize',20);
% %zlabel('$\mathbf{u}_{\rm ode45}$','interpreter','latex','fontsize',20);
% xlim([0,t(nt)]);
% zlabel('$\mathbf{u}_{\rm ParaDIAG-I}$','interpreter','latex','fontsize',20);
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function val=StepAC(V,F,Nx)
global Nt
G=zeros(Nx*Nt,1);
for j=1:Nt
Z=zeros(Nx,1);
for k=1:Nt
Z=Z+V(j,k)*F((k-1)*Nx+1:k*Nx,1);
end
G((j-1)*Nx+1:j*Nx,1)=Z;
end
val=G;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [V,D,invV]=getV(dtn)
global tau Nt;
p=zeros(Nt-1,1);
q=zeros(Nt-1,1);
tD=ones(Nt,1);
invtD=ones(Nt,1);
for n=1:Nt-1
p(n)=prod((1+tau.^(1:n))./(1-tau.^(1:n)));
q(n)=(1/tau^n)*prod((1+(tau^2)*tau.^(-n:-1))./(1-tau.^(-n:-1)));
tD(n)=1/sqrt(1+sum(abs(p(1:Nt-n)).^2));
invtD(n)=sqrt(1+sum(abs(p(1:Nt-n)).^2));
end
V=toeplitz([1;p],[1,zeros(1,Nt-1)]);
invV=toeplitz([1;q],[1,zeros(1,Nt-1)]);
D=2./dtn;
end