use successive over relaxation method to solve finite-difference form of the laplacian operator in matlab

To use the Successive Over Relaxation (SOR) method to solve the finite-difference form of the Laplacian operator in MATLAB, you can follow the steps below:

Set up the problem:
- Define the domain and grid points.
- Specify the boundary conditions.
- Initialize the solution matrix.
Define the SOR solver function:
- Iterate over the grid points until convergence is reached.
- At each point, update the solution based on neighboring points.
- Use the over relaxation factor (w) to control convergence speed.
Implement the SOR method in MATLAB:
- Define a function that takes the domain, boundary conditions, and SOR parameters as input.
- Inside the function, set up a loop to iterate over the grid points until convergence.
- Update the solution matrix based on the SOR update formula.

Here's an example MATLAB code that demonstrates the usage of the SOR method for solving the finite-difference Laplacian operator:


main.m
function u = solveLaplacianSOR(domain, boundary, w, tol)
    % domain: matrix representing the domain and grid points
    % boundary: matrix representing the boundary conditions
    % w: over relaxation factor
    % tol: convergence tolerance
    
    % Extract the dimensions of the domain
    [m, n] = size(domain);
    
    % Initialize the solution matrix
    u = domain;
    
    % Set up convergence check variables
    converged = false;
    iter = 0;
    
    % Iterate until convergence is reached
    while ~converged
        % Initialize the maximum residual
        maxResidual = 0;
        
        % Iterate over the grid points
        for i = 2:m-1
            for j = 2:n-1
                % Check if the current point is not on the boundary
                if ~boundary(i, j)
                    % Calculate the SOR update
                    new_val = (1 - w) * u(i, j) + (w / 4) * (u(i+1, j) + u(i-1, j) + u(i, j+1) + u(i, j-1));
                    
                    % Calculate the residual at the current point
                    residual = abs(new_val - u(i, j));
                    
                    % Update the solution and maximum residual
                    u(i, j) = new_val;
                    maxResidual = max(maxResidual, residual);
                end
            end
        end
        
        % Check for convergence
        if maxResidual < tol
            converged = true;
        end
        
        % Increment the iteration count
        iter = iter + 1;
    end
    
    % Display the convergence information
    disp(['Converged in ', num2str(iter), ' iterations.']);
end
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You can call the solveLaplacianSOR function by passing the domain, boundary conditions, over relaxation factor (w), and convergence tolerance (tol) as inputs. After convergence, the function will return the solution matrix u.


main.m
% Example usage
% Define the domain and boundary conditions
domain = zeros(5, 5);  % for simplicity, assuming a 5x5 grid
boundary = false(size(domain));
boundary(1, :) = true;
boundary(end, :) = true;
boundary(:, 1) = true;
boundary(:, end) = true;
domain(3, 3) = 1;  % randomly initialize a point in the domain

% Solve using SOR method
w = 1.5;  % over relaxation factor
tol = 1e-6;  % convergence tolerance
u = solveLaplacianSOR(domain, boundary, w, tol);
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Note that the convergence of the SOR method depends on the choice of the over relaxation factor (w) and the grid spacing. Experiment with different values of w and watch the convergence behavior.

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