I need programming to solve wave equation using finite difference method

d^2u/dx^2=c^2 d^2u/dt^2

Answer 1

[SAI SRUJAN]

Hi Kasim,

I understand that you are trying to solve wave equation using finite difference method.

The CFL condition essentially dictates the relationship between the time step size ((dt)), the spatial step size ((dx)), and the wave speed ((c)) for a stable numerical simulation.Refer to the following code sample ensuring the CFL condition is met and proceed further,

% Parameters
L = 1.0; % Length of the domain
T = 1.0; % Total time
c = 1.0; % Wave speed

nx = 100; 
nt = 100;
dx = L / (nx - 1);
dt = T / (nt - 1);

x = linspace(0, L, nx);
u = zeros(nx, nt);

% Ensure CFL condition is satisfied
if c * dt / dx > 1
    error('CFL condition not satisfied: c * dt / dx must be <= 1');
end

% Initial conditions - Adjust accordingly.
u(:, 1) = sin(pi * x); % Initial displacement
u(:, 2) = u(:, 1); % Assume initial velocity is zero

% Finite difference method
for t = 2:nt-1
    for i = 2:nx-1
        u(i, t+1) = 2 * (1 - (c * dt / dx)^2) * u(i, t) - u(i, t-1) + ...
            (c * dt / dx)^2 * (u(i+1, t) + u(i-1, t));
    end
end

% Visualization
figure;
for t = 1:nt
    plot(x, u(:, t));
    ylim([-1, 1]);
    title(['Time step: ', num2str(t)]);
    xlabel('x');
    ylabel('u(x, t)');
    drawnow;
    pause(0.05);
end

I hope this helps!