I know that forward euler is probably not the most accurate approach for discretizing a PDE. However, due to its simplicity, it can be a good starting step.
It is easy to code, however, it is also prone to issues of stability. Care must be taken to choose the time step size and the mesh width. For many famous PDEs like the heat equation, such conditions are well known and taught in numerical analysis classes.
However, for less-known PDEs, especially for nonlinear ones, I am unsure how one would proceed to do a stability analysis of explicit finite difference schemes like forward Euler. von Neumann stability analysis, according to Wikipedia, is only for linear PDEs. Can anyone point me to any references -- books or lecture notes -- that illustrate how stability analysis can be done for nonlinear PDEs?
My main purpose for asking this is perhaps it would facilitate my computer implementation of the explicit scheme for a particular nonlinear PDE. I could always try various time steps and mesh widths and continuously refine them and observe whether the solution has "converged". But is there another more efficient way about it?
Thanks!