Example 6.4

Poisson Equation with non Homogeneous Boundary Conditions

Contents

Initialization
Define the RHS of modified eq
Solve using DST
Solution of the original equation
Generates the plots

Initialization

close all;
clear M N dx dy x y X Y Pji Pij Pjk fun Qij Qkj Qjk
clear k A b Sol

M = 32;
N = 32;
dx= 1/M;
dy= 1/N;
x = [0:dx:1];
y = [0:dy:1];
[X,Y] = meshgrid(x,y);

Pji = zeros(N+1,M+1);
Pij = zeros(M+1,N+1);
Pjk = zeros(M+1,N+1);

Define the RHS of modified eq

fun = @(x,y) 30*(x.^2-x) + 30*(y.^2-y) - 4*pi^2*(x-1).*sin(2*pi*y);
Qij = fun(X,Y);
Qkj = dst1(Qij');
Qjk = Qkj';

Solve using DST

for k = [1:M-1]
    %Solve the tridiagonal system for each k
    A = diag(ones(M-2,1),-1) + diag(ones(M-2,1),1) + ...
        diag((((dy/dx)^2*(2*cos(pi*k/M)-2))-2)*ones(M-1,1));
    A = sparse(A);
    b = dy^2 * Qjk(2:M,k+1);
    Pjk(2:M,k+1) = (A^-1)*b;
end;

%Inverse transform
Pji = dst1(Pjk')*M/2;
Pij = Pji';

Solution of the original equation

Sol = Pij - (X-1).*sin(2*pi*Y);

Generates the plots

surf(X,Y,Sol);
xlabel('x');ylabel('y');zlabel('phi');
title('Numerical solution of Poisson equation, ex 6.4');