I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation.
I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve a nonlinear, time-dependent, heat flow (thermal diffusion) equation in cylindrical symmetry. As such, I am interested in the temperature as a function of time t, radius R, and axial position Z (no $\theta$ dependence).
In my case, the radius is larger than the thickness (R >> Z), roughly 10 to 1.
My question is this: when deciding the spacing (or stencil?) between points/nodes for my simulation, is it better to:
(1) Use the same number of points in each spatial dimension (say, 100 in R, and 100 in Z)?
or,
(2) Use the same ratio of points at the ratio of the spatial dimensions (say, 20 in R and 2 in Z, i.e. 10 to 1)?
As I said, I'd like to know the proper way to do it for general cases, but knowing for my case would be helpful by itself.
Also, does the time step also need to obey some equality in comparison to the spacing in R and Z at the same time? I've seen conditions for convergence, but those appear to be some ratio between time and one spatial dimension, not time and all spatial dimensions. Thanks!
My understanding is (IMHO), that it is more important that the spacing (or in your case, or ratio of points) be equal in each dimension. That is to say, for your problem, that dr = dz, otherwise, diffusion will occur at different rates in different directions, leading to "false diffusion". I don't have a source or reference for this. So if R>>Z (R:Z = 10:1), then for dr to equal dz, you would need 10 times as many points in the r-direction than the z-direction. For Cartesian this is more straight forward because the size of your cells remains constant if you take this approach, however with cylindrical (and spherical ?), it is a little more difficult to see.
If someone else wants to weigh in on this and/or has a proof or reference, I'd like to know!