stability comparison
 

I have used two method to solve a system of nonlinear equation, with the same number of time steps, mesh, and solver, etc. Can I comment on the stability of the two methods, based on the largest time step sizes that still produces stable results? (I know it will not be a definite proper stability analysis) - any other suggestions are also welcome.

Also, if I the results seem stable, but the CFL is no longer satisfied with the larger time steps, it is correct to take the step size at which CFL condition is not satisfied as the largest step size, even though the results may seem stable?

This approach is crude, but I do think it can work.

I am skeptical that you actually know beforehand the exact CFL condition for a problem in general (especially nonlinear ones). The CFL condition is generally a necessary but insufficient condition. Violation of the CFL condition means that the problem is unstable. However, the CFL condition(s) are actually specific to each problem and scheme. If you have the prowess to prove it, it is helpful to derive the CFL condition for your problem.

Flux inhibitors and gradient limiters also help improve stability, by a lot, and this generally isn't accounted for. Both physical & numerical diffusion also improve stability.

generally, for a convection-diffusion 3d problem, the numerical stability implies the existence of a region of stability...that is a hypersurface being cx,cy,cz, alphax,alphay,alphaz the numerical parameters..
and the Neumann analysis gives a response only for the linear case..

Thank you LuckyTran. Would another (crude) approach be to compare the time the calculations take to reach convergence?