I have solved a numerical problem using two different sets of non-linear governing equations. I want to get an understanding of the stability of the methods relative to each other. To do so, I solving the problem with each method whilst keeping the time stepping schemes, time step and mesh sizes the same. I then run the code, and observe the largest time step using which stable results are produced or convergence is obtained. I wanted to know whether this makes sense. I know this process does not get definite answer about stability of the methods.
Also what I have found that for this particular problem, stable results are still obtained even with CFL numbers greater than 1 (very large time steps). Do the large time steps I try still need to satisfy CFL?