When I was comparing the Laplacian kernels of 5-point and 9-point, I found that the 9-point kernel (such as the kernel with a value of 8 in the center and -1 in all 8 neighboring points) simply added the four values at the diagonal vertices as values from the top, bottom, left, and right. But isn't the distance between the vertex of the diagonal and the center point square root of 2 right? So, how can we treat them as “h=1” in the 5-point kernel for finite difference?
Anyone please provide me with a detailed derivation process for the 9-point stencil without omit some steps.
Thanks a lot!