advection diffusion solution in matlab

main.m
% Define parameters and initial conditions
D = 0.1; % diffusion coefficient
v = 0.2; % advection velocity
L = 1; % domain length
T = 1; % total time 
dx = 0.01; % spatial step size
dt = 0.001; % time step size
x = 0:dx:L; % domain
t = 0:dt:T; % time domain
N = length(x); % number of spatial points
M = length(t); % number of time points
u = zeros(N,M); % initialize solution array
u(:,1) = sin(pi.*x); % initial condition

% Apply finite difference method to solve advection-diffusion equation
for n=2:M
    u(2:N-1,n) = u(2:N-1,n-1) + ... % diffusion term
        D*dt/(dx^2)*(u(3:N,n-1)-2*u(2:N-1,n-1)+u(1:N-2,n-1)) - ... 
        v*dt/(2*dx)*(u(3:N,n-1)-u(1:N-2,n-1)); % advection term
    u(1,n) = u(2,n); % boundary condition at x = 0
    u(N,n) = u(N-1,n); % boundary condition at x = L
end

% Plot solution
[X,T] = meshgrid(x,t);
surf(T,X,u);
xlabel('Time');
ylabel('Position');
zlabel('Solution');
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