Just for the sake of somebody else coming here for the same problem. The usual way to convert the Laplacian to a finite difference expression on a regular grid is:
∆u(x,y) -> idx2*[u(x+1,y) + u(x-1,y) - 2*u(x,y)] +
idy2*[u(x,y+1) + u(x,y-1) - 2*u(x,y)]
where idx2
and idy2
are the inverse squares of the grid spacing in dimension x and y respectively. In the case when the grid spacing in both dimensions is the same, this simplifies to:
∆u(x,y) -> igs2*[u(x+1,y) + u(x-1,y) + u(x,y+1) + u(x,y-1) - 4*u(x,y)]
The multiplicative coefficient can be removed by hiding it inside other coefficients, e.g. c
, by changing their units of measurement:
∆u(x,y) -> u(x+1,y) + u(x-1,y) + u(x,y+1) + u(x,y-1) - 4*u(x,y)
By the way, there cannot be 2D spherical waves since spheres are 3D objects. 2D waves are called circular waves.