The pseudo-spectral collocation method is a numerical method for solving differential equations using the spectral representation of the solution. In this method, the solution is represented as a sum of basis functions (e.g., Chebyshev or Fourier polynomials), and the collocation points are used to enforce the equation's constraints. I will give you a simple example of how to implement the Chebyshev collocation method in MATLAB to solve the Poisson equation on the interval [-1, 1]:
Poisson equation: -u''(x) = f(x) with u(-1) = u(1) = 0.
Let f(x) = 10*sin(3*pi*x). The analytical solution is u(x) = (10/(9*pi^2)) * sin(3*pi*x).
This is a MATLAB implementation of the Chebyshev collocation method to solve this problem:
% Problem definition
f = @(x) 10 * sin(3 * pi * x);
analytical_solution = @(x) (10 / (9 * pi^2)) * sin(3 * pi * x);
N = 20; % Number of collocation points
% Chebyshev collocation points
x = cos(pi * (0:N) / N)';
% Chebyshev differentiation matrix
T = chebyshev_differentiation_matrix(N);
% Second-order differentiation matrix
T2 = T^2;
% Boundary conditions
T2 = T2(2:end-1, 2:end-1);
% Evaluate the forcing term at the collocation points
F = f(x);
F = F(2:end-1);
% Solve the linear system
u_inner = T2 \ F;
% Add boundary conditions back
u = [0; u_inner; 0];
% Plot the numerical and analytical solutions
xx = linspace(-1, 1, 200);
figure;
plot(x, u, 'ro', xx, analytical_solution(xx), 'b-');
legend('Numerical solution', 'Analytical solution');
xlabel('x'); ylabel('u(x)');
title('Chebyshev collocation method for Poisson equation');
In this example, we first define the problem and the number of collocation points N. Then, we calculate the Chebyshev collocation points and the Chebyshev differentiation matrix using a custom helper function chebyshev_differentiation_matrix. We then compute the second-order differentiation matrix and apply the boundary conditions. Finally, we solve the linear system and plot the numerical and analytical solutions.
Here is the chebyshev_differentiation_matrix function:
function D = chebyshev_differentiation_matrix(N)
% Compute the Chebyshev differentiation matrix of order N
x = cos(pi * (0:N) / N)';
c = [2; ones(N-1, 1); 2] .* (-1).^(0:N)';
X = repmat(x, 1, N+1);
dX = X - X';
D = (c * (1./c)')./(dX + eye(N+1));
D = D - diag(sum(D, 2));
end
This is how the output looks like:
