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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,12 @@ | ||
| function [x, w] = gaulob(a, b, n) | ||
| % Returns abscissas and weights for the Gauss-Lobatto n-point quadrature | ||
| % over the interval [a, b] using the Golub-Welsch Algorithm. | ||
| x=zeros(1,n); x([1,n])=[-1,1]; | ||
| w=zeros(1,n); w([1,n])=(b-a)/(n*(n-1)); | ||
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| k=1:n-3; | ||
| E=sqrt((k.*(k+2))./((2*k+1).*(2*k+3))); | ||
| [x(2:n-1),V]=trideigs(zeros(1,n-2), E); | ||
| w(2:n-1)=2/3*(b-a)*V(1,:).^2./(1-x(2:n-1).^2); | ||
| x=(b-a)/2*x+(a+b)/2; | ||
| end |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,93 @@ | ||
| function [] = brillTest(m,n) | ||
| n(1:length(m))=n; | ||
| err=zeros(size(m)); | ||
| millis=zeros(numel(m),2); | ||
| for i=1:length(m) | ||
| [err(i), millis(i,:)]=brillSolve(m(i),n(i)); | ||
| drawnow; | ||
| end | ||
| figure(2); | ||
| semilogy(m,err); | ||
| title('Error |\Delta|_\infty'); | ||
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| figure(3); | ||
| plot(m,millis); | ||
| title('Time (ms)'); | ||
| end | ||
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| function [err,millis] = brillSolve(m,n) | ||
| millis=[0,0]; | ||
| tic; | ||
| % Simulation parameters | ||
| L=8; | ||
| r0=sqrt(2)*L; | ||
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| [Dx,x]=chebD(2*m); | ||
| A1=diag(x.^2)*Dx*Dx+diag(2*x)*Dx; | ||
| [A1,Dx,x]=radial(A1,Dx,x); | ||
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| [Dy,y]=chebD(n); y=y'; | ||
| A2=diag(1-y.^2)*Dy*Dy-diag(2*y)*Dy; | ||
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| a=[1,1;0,0]; | ||
| b=[1,1;1,1]; | ||
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| % Imposition of boundary conditions | ||
| E1=eye(m); | ||
| E2=eye(n); | ||
| B1=a(1,1)*E1(1,:)+b(1,1)*Dx(1,:); | ||
| B2=diag(a(2,:))*E2([1,end],:)+diag(b(2,:))*Dy([1,end],:); | ||
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| b1=0*y+1.000068252008242; | ||
| b2=[0*x,0*x]; | ||
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| % Coordinate mapping | ||
| r=r0*x; | ||
| th=acos(y); | ||
| rho=r*sin(th); | ||
| z=r*cos(th); | ||
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| % Test code | ||
| C=(1/2)*exp(-rho.^2-z.^2).*(1+2*rho.^2.*(-3+rho.^2+z.^2)); | ||
| F=(1/20).*(1+rho.^2+z.^2).^(-5/2).*((-6)+exp(-rho.^2-z.^2) ... | ||
| .*(1+rho.^2+z.^2).^2.*(1+2.*rho.^2.*((-3)+rho.^2+z.^2)).*( ... | ||
| 1+10.*(1+rho.^2+z.^2).^(1/2))); | ||
| C=diag(r.^2)*C; | ||
| F=diag(r.^2)*F; | ||
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| % Solution | ||
| [green,ps,kd,sc,gb]=elliptic(A1,A2,B1,B2,1,[1,n]); | ||
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| ub=ps(b1,b2); | ||
| rhs=kd(F-A1*ub-ub*A2'-C.*ub); | ||
| u0=kd(ub+green(rhs)); | ||
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| afun=@(uu) (sc(uu)+kd(C).*uu); | ||
| pfun=@(uu) kd(green(uu)); | ||
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| millis(1)=toc; | ||
| tic; | ||
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| [u1,res,its]=precond(afun,pfun,rhs,u0,20,2e-16); | ||
| uu=gb(u1)+ub; | ||
| millis(2)=1000*toc; | ||
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| ug=1+(1/10).*(1+rho.^2+z.^2).^(-1/2); | ||
| err=norm(kd(uu-ug),'inf'); | ||
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| display(its); | ||
| display(res); | ||
| display(err); | ||
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| figure(1); | ||
| surf(kron([-1,1],rho), z(:,[end:-1:1,1:end]), uu(:,[end:-1:1,1:end])); | ||
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| colormap(jet(256)); | ||
| colorbar; | ||
| shading interp; | ||
| %camlight; | ||
| axis square; | ||
| xlim([-L,L]); | ||
| ylim([-L,L]); | ||
| view(2); | ||
| end |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,6 +1,8 @@ | ||
| function [uu,res,its] = precond(eqn, green, rhs, u0, maxit, tol) | ||
| % bicgstab helper for 2D elliptic equations | ||
| afun=@(x) reshape(eqn(reshape(x, size(u0))), [], 1); | ||
| pfun=@(x) reshape(green(reshape(x, size(u0))), [], 1); | ||
| [x,~,res,its]=bicgstab(afun,rhs(:),tol,maxit,pfun,[],u0(:)); | ||
| %[x,~,res,its]=gmres(afun,rhs(:),[],tol,maxit,pfun,[],u0(:)); | ||
| uu=reshape(x,size(u0)); | ||
| end |
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