i have a code for solving the stream function using successive line over relaxation , i should get a graph as metioned below can anyone do correcction tions in this code

nx = 41;

ny = 41;

Lx = 1.0;

Ly = 1.0;

A = Lx;

dx = Lx / (nx - 1);

dy = Ly / (ny - 1);

omega = 0.5;

tol = 1e-5;

max_iter = 5000;

% Initialize Psi, zeta, and temperature T

Psi = zeros(nx, ny); % Initial guess for Psi (stream function)

zeta = zeros(nx, ny); % Initial guess for vorticity

T = zeros(nx, ny); % Temperature

% Boundary conditions for initial time tau = 0

T(:, :) = 0; % Initially, T = 0 throughout

zeta(:, :) = 0; % Initially, zeta = 0 throughout

% Boundary conditions for tau > 0

% X = 0 and X = A (left and right boundaries)

Psi(:, 1) = 0; % Psi = 0 at X = 0

Psi(:, end) = 0; % Psi = 0 at X = A

% dPsi/dX = 0 at X = 0 and X = A (already satisfied as Psi = 0 at boundaries)

% dT/dX = 0 at X = 0 and X = A (no-flux boundary)

% Y = 0 (bottom boundary)

Psi(1, :) = 0; % dPsi/dY = 0 at Y = 0

T(1, :) = -1; % T = -1 at Y = 0

% Y = 1 (top boundary)

Psi(end, :) = 0; % dPsi/dY = 0 at Y = 1

T(end, :) = 1; % T = +1 at Y = 1

% Main SLOR iteration loop

for iter = 1:max_iter

Psi_old = Psi; % Save old Psi for comparison

for i = 2:nx-1

for j = 2:ny-1

% Compute the velocity components U and V using centered differences

U = (Psi(i, j+1) - Psi(i, j-1)) / (2 * dy); % U = dPsi/dY

V = -(Psi(i+1, j) - Psi(i-1, j)) / (2 * dx); % V = -dPsi/dX

% Successive Line Over Relaxation formula for Psi

Psi(i, j) = (1 - omega) * Psi(i, j) + ...

omega / 4 * (Psi(i+1, j) + Psi(i-1, j) + ...

Psi(i, j+1) + Psi(i, j-1) + dx^2 * zeta(i, j));

end

end

% Compute the vorticity zeta using centered differences after Psi update

for i = 2:nx-1

for j = 2:ny-1

% zeta = dV/dX - dU/dY

zeta(i, j) = ((Psi(i+1, j) - Psi(i-1, j)) / (2 * dx)) - ...

((Psi(i, j+1) - Psi(i, j-1)) / (2 * dy));

end

end

% Check for convergence using the relative error

error = max(max(abs(Psi - Psi_old)));

if error < tol

fprintf('Converged after %d iterations with error %e\n', iter, error);

break;

end

end

% If maximum iterations are reached without convergence

if iter == max_iter

disp('Maximum iterations reached without convergence.');

end

% Plotting the stream function Psi

figure;

contourf(linspace(0, Lx, nx), linspace(0, Ly, ny), Psi', 20);

colorbar;

title('Stream Function \Psi using SLOR');

xlabel('x');

ylabel('y');

Answer 1

[Malay Agarwal]

Hi @Ruthra,

The problem is with the initial guess for Ψ. Since you initialize it as zeros, the following equation:

(1 - omega) * Psi(i, j) + ...

omega / 4 * (Psi(i+1, j) + Psi(i-1, j) + ...

Psi(i, j+1) + Psi(i, j-1) + dx^2 * zeta(i, j))

Always yields zero. Thus, Psi is never updated and the SOR converges after a single iteration with all values in Psi set to zero. This leads to the contour plot not being as expected. I am not sure how Psi should be initialized but if I just do random initialization (refer to the attached file), I do get something sensible:

sor

Converged after 1418 iterations with error 9.967857e-06

Note that I have also changed the number of levels to 7 since that's how many I counted in your expected plot.

Hope this helps!

[Ruthra]

may i know what initial guess you have given for psi

[Malay Agarwal]

I have initialized Psi randomly.

Psi = rand(nx, ny);

You can refer to the attached file for the full code.

[Ruthra]

Your answer is so helpful,there is no attached file.Can you plese send the code you have run once again .Thank you

[Malay Agarwal]

Please check at the top of the answer, just above the "Hi". There's a file called "sor.m".