Solve 2D PDE problem by Finite Difference Method

I need to solve a 2D PDE problem:

 a* dT/dt + b* dT/dx = c * d2T/dz2

To generate the temperature at each position at different time by finite difference method (I know the equations of solving dT/dt, dT/dx, dT/dz, and d2T/dz2)

But I don't know how to program and plot this..

Thank you~

Answer 1

[SAI SRUJAN]

Hi Howard,

I understand that you are facing an issue in solving the 2D PDE problem by finite difference method.

Refer to the following code sample to proceed further,

% Parameters Update parameters as required
a = 1;  
b = 1;  
c = 1;  

dx = 0.1; % Spatial step in x
dz = 0.1; % Spatial step in z
dt = 0.01; % Time step
x = 0:dx:1; 
z = 0:dz:1; 
t = 0:dt:1; 

Nx = length(x);
Nz = length(z);
Nt = length(t);

% Initial condition
T = zeros(Nx, Nz);
T(:, :) = 100; % Example initial temperature

% Boundary conditions - Update boundary conditions as required
T(:, 1) = 0; % Boundary at z=0
T(:, end) = 0; % Boundary at z=end

% Finite Difference Coefficients
alpha = c * dt / dz^2;
beta = b * dt / dx;

for n = 1:Nt-1
    T_new = T;
    for i = 2:Nx-1
        for j = 2:Nz-1
            % Update temperature using finite difference method
            T_new(i, j) = T(i, j) + ...
                          alpha * (T(i, j+1) - 2*T(i, j) + T(i, j-1)) - ...
                          beta * (T(i+1, j) - T(i, j));
        end
    end
    T = T_new;
    
    % Plotting the temperature distribution
    surf(x, z, T');
    title(['Temperature distribution at time t = ', num2str(t(n))]);
    xlabel('x');
    ylabel('z');
    zlabel('Temperature');
    drawnow;
end

I hope this helps!