On 23/06/16 22:06, Slinker, Kyle Patrick wrote:
Thanks for the reply, Eloisa.
I wasn't thinking of solving the elliptic equation in terms of the stationary state of a parabolic equation as you described in your paper. But, I see now how Gauss-Seidel for the elliptic equation can be derived from finite differencing the parabolic equation. Now that I think I'm on the same page in those terms, let me see if I can rephrase the issue I'm seeing.
I tried a couple times to write something, but the best explanation I came up with is an example. I attached a short PDF walking through it.
Thanks again for your help.
Dear Kyle,
I've followed your reasoning, but I don't see where equation (2) comes from. To the best of my knowledge, one is not free to construct an iterative process by deforming the differencing stencils at will. The existing recipes (like Gauss-Seidel) are carefully crafted to have specific properties; you can, for instance, read on the Numerical Recipes book (equation 20.5.4 and following, in the third edition) what dtime needs to be set to for a stable evolution. This has to do with the stability of the Forward-Time-Centered-Space representation of the equation.
Notice, however, that what dtime is set to in CT_MultiLevel is only the largest admissible value. One is free to decrease this number (although that would require more iterations to relax to the same state); your suggestion for the coefficient, for instance, would also work. And you are right to point out that a change in dtime is equivalent to a change in the SOR omega (which, however, also cannot be chosen arbitrarily).
I hope this clarifies the issue!
Eloisa