Matrix to generate finite difference

Question

I'm implementing a finite difference scheme for a 2D PDE problem. I wish to avoid using a loop to generate the finite differences. For instance to generate a 2nd order central difference of u(x,y)_xx, I can multiply u(x,y) by the following:

Is there a nice matrix representation for u_xy = (u_{i+1,j+1} + u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1})/(4dxdy)? It's a harder problem to code as it's in 2D - I'd like to multiply some matrix by u(x,y) to avoid looping. Many thanks!

Answer 1

If your points are stored in a N-by-N matrix then, as you said, left multiplying by your finite difference matrix gives an approximation to the second derivative with respect to u_{xx}. Right-multiplying by the transpose of the finite difference matrix is equivalent to an approximation u_{yy}. You can get an approximation to the mixed derivative u_{xy} by left-multiplying and right-multiplying by e.g. a central difference matrix

delta_2x =

     0     1     0     0     0
    -1     0     1     0     0
     0    -1     0     1     0
     0     0    -1     0     1
     0     0     0    -1     0

(then divide by the factor 4*Dx*Dy), so something like

U_xy = 1/(4*Dx*Dy) * delta_2x * U_matrix * delta_2x';

If you cast a matrix as a N^2 vector

U_vec = U_matrix(:);

then these operators can be expressed using a Kronecker product, implemented in MATLAB as kron: We have

A*X*B = kron(B',A)*X(:);

so for your finite difference matrices

U_xy_vec = 1/(4*Dx*Dy)*(kron(delta_2x,delta_2x)*U_vec);

If instead you have an N-by-M matrix U_mat, then left matrix multiplication is equivalent to kron(eye(M),delta_2x_N) and right multiplication to kron(delta_2y_M,eye(N)), where delta_2y_M (delta_2x_N) is the M-by-M (N-by-N) central difference matrix, so the operation is

U_xy_vec = 1/(4*Dx*Dy) * kron(delta_2y_M,delta_2y_N)*U_vec;

Here is an MATLAB code example:

N = 20;
M = 30;
Dx = 1/N;
Dy = 1/M;

[Y,X] = meshgrid((1:(M))./(M+1),(1:(N))/(N+1));

% Example solution and mixed derivative (chosen for 0 BCs)
U_mat = sin(2*pi*X).*(sin(2*pi*Y.^2));
U_xy = 8*pi^2*Y.*cos(2*pi*X).*cos(2*pi*Y.^2);

% Centred finite difference matrices
delta_x_N = 1/(2*Dx)*(diag(ones(N-1,1),1) - diag(ones(N-1,1),-1));
delta_y_M = 1/(2*Dy)*(diag(ones(M-1,1),1) - diag(ones(M-1,1),-1));

% Cast U as a vector
U_vec = U_mat(:);

% Mixed derivative operator
A = kron(delta_y_M,delta_x_N);

U_xy_num = A*U_vec;
U_xy_matrix = reshape(U_xy_num,N,M);

subplot(1,2,1)
contourf(X,Y,U_xy_matrix)
colorbar
title 'Numeric U_{xy}'
subplot(1,2,2)
contourf(X,Y,U_xy)
colorbar
title 'Analytic U_{xy}'