Speeding up 2D Finite Difference Matrix

I have two matrices B and D, where B is a matrix and D is a matrix defined as follows:

with ,

where is the identity matrix and denotes the Kronecker product of matrices P and Q. I am currently computing B as a sparse matrix as follows:

o = ones(n,1);
z = zeros(n,1);
D = spdiags([-o o z],-1:1,n,n);
D(1,1) = 0;

I = speye(n);

B1 = kron(I, D);

B2 = kron(D, I);

B = [B1;B2];

I know that can be quickly computed using

[0; diff(x)]

I need to find a quick way of computing and in a similar manor. Currently my code is spending a signifigant amount of its time to compute and so I was wondering if anyone had some incite into computing this efficently.

Answer 1

[Chris J]

I solved it here is my solution:

U = reshape(x, [n,n]);
u1 = [zeros([1,n]); diff(U, 1, 1)];
u2 = [zeros(n, 1), diff(U, 1, 2)];
y = [u1(:); u2(:)];

To solved for we solve the equavalent formulation :

x1vec = x(1:end/2);
x2vec = x(end/2+1:end);
    
x1mat = reshape(x1vec, [n,n]);
x2mat = reshape(x2vec, [n,n]);

p1 = -diff(x1mat, 1, 1);
p1 = [-x1mat(2, :); p1(2:end,:); x1mat(end,:)];
p1 = p1(:);

p2 = -diff(x2mat, 1, 2);
p2 = [-x2mat(:,2), p2(:, 2:end), x2mat(:,end)];
p2 = p2(:);

y = p1+p2;