In reading Nick Trefethen's Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström and Group velocity in finite difference schemes, the thought occurred to me that if instability is a result of reflections at boundaries, can the theory be extended to meshes? This is somewhat analogous to reading a wave-theoretical exposition of Snell's Law and asking if the theory can be extended to anisotropic inhomogeneous media. I suspect that it can, but I don't know if it's worthwhile.
Now, the velocity of propogation of light in a medium is determined by the permittivity and permeability of that medium. From these quantities, we can also calculate the characteristic impedance of the medium. We generally think of media as linear, isotropic, and homogeneous, but this is not the case in general. We often think of monochromatic light but, again, this is not the case in general. When light passes from one medium to another, a difference in the characteristic impedance of the media leads to reflections at the boundary.
What Trefethen started me thinking was, is there a kind of “permittivity” of a grid and a “permeability” for a method which would give a wavenumber-dependent “characteristic impedance” for the grid/method combination and a different characteristic impedance for boundaries, owing to either imposed or numerically necessary boundary conditions, and allowing instability to be determined easily?
Generalising to an irregular mesh, the mesh analogue of permittivity would be a wavenumber-dependent discrete tensor. However intimidating it might sound, and however intractable it might be for a human to calculate, it would surely allow a computer to predict, given a mesh and a method, what Fourier modes would cause instability and, perhaps more importantly, where in the mesh this might arise. A potential end-product might be a software tool which takes a mesh from, say, COMSOL Multiphysics, and identifies “hot spots” (analogous to “shiny lumps” in an optical medium) where particular Fourier modes would reflect, giving rise to instability. Given the Fourier transform of a particular initial condition or forcing function, instability might be predicted and avoided by mesh refinement.
Now, it may be that this analysis would end up being equivalent to solving the problem on the mesh in full generality, or be obscenely computationally costly for some other reason, in which case, I admit, the entire idea is utterly worthless, but there's no harm in asking, right?