[jist]

Hi all,
I've been given analytical solution of U=x^2-y^2 which satisfies the Laplace Eq. and its numerical solution as S for a square domain. Boundary conditions setted from analytical solution 'U' and unchanged through the code. Internal points of S taken as 0 at initial.
Then i am asked that to perform 20 iterations of SOR method with relaxation factor (omega) changing 1 to 2 with an interval of 0.002.
I should've been able to calculate error between analytical and numerical solution for each relaxation factor at the end of 20th iteration with given MATLAB code below but errors do not match with the solution at all.
Can anybody tell me what is it that i am doing wrong? Thanks in advence.
Here is my code,

clc
close all
clear all

N=21;
iteration=20;
deltax=1/(N-1);
deltay=deltax;
delomega=0.002;
omegai=1;
Nomega=1/(0.002);
x=linspace(0,1,21);
y=linspace(0,1,21);
omega=linspace(1,2,Nomega);

%analytical solution 
u=zeros(N,N);
for j=1:N
    for i=1:N
        u(i,j)=x(i)^2-y(j)^2;
    end
end
s=zeros(N,N);
s(1,:)=u(1,:);
s(N,:)=u(N,:);
s(:,1)=u(:,1);
s(:,N)=u(:,N);

error=0;
for k=1:Nomega
    for l=1:iteration
        for j=2:N-1
            for i=2:N-1
                R=s(i+1,j)+s(i-1,j)+s(i,j+1)+s(i,j-1)-4*(s(i,j))-(deltax^2)*s(i,j); 
                s(i,j)=s(i,j)+0.25*(1+delomega*k)*R;
            end
        end
    if l==iteration
        error=error+abs(u(i,j)-s(i,j));
        %error_(k,1)=omega(k);
        error_(k)=error/((N-2)*(N-2));
    end
    end
end

plot(omega(:),error_(:));
xlabel('\omega');
ylabel('Error');
grid on
grid minor

[FMDenaro]

First, correct "U=x^2+y^2" in "U=x^2-y^2", otherwise the solution does not satisfy the Laplace euqation.


Then, why do you think to get an accurate solution after only 20 iterations??

[jist]

Sorry for the typo, it was x^2-y^2 as stated in the code.
Well this part is not about how to get an accurate solution via iteration. I will simply run the code for smaller mesh sizes to see how mesh sizes impact the solution.

Originally Posted by FMDenaro

First, correct "U=x^2+y^2" in "U=x^2-y^2", otherwise the solution does not satisfy the Laplace euqation.


Then, why do you think to get an accurate solution after only 20 iterations??

[FMDenaro]

Originally Posted by jist

Sorry for the typo, it was x^2-y^2 as stated in the code.
Well this part is not about how to get an accurate solution via iteration. I will simply run the code for smaller mesh sizes to see how mesh sizes impact the solution.

You wrote "but errors do not match with the solution at all."

[jist]

I have the solution plot (error as a function of omega and grid sizes) for the scenario i explained but i am not getting the same results. What i tried to ask was that am i doing anything wrong with SOR or the error between analytical and numerical solution?

Originally Posted by FMDenaro

You wrote "but errors do not match with the solution at all."

[FMDenaro]

You have a 2D map of the error as function of two variables. Plot the contour and the surface and check for the values producing the minimum error. Be aware that you cannot see a decreasing of the error for smaller mesh sizes.