A very common adage in ODE solvers is that if you run into trouble with an explicit method, usually some explicit Runge-Kutta method like RK4, then you should try an implicit method. Implicit methods, because they are doing more work, solving an implicit system via a Newton method having “better” stability, should be the thing you go to on the “hard” problems.

This is at least what I heard at first, and then I learned about edge cases. Specifically, you hear people say “but for hyperbolic PDEs you need to use explicit methods”. You might even intuit from this “PDEs can have special properties, so sometimes special things can happen with PDEs… but ODEs, that should use implicit methods if you need more robustness”. This turns out to not be true, and really understanding the ODEs will help us understand better … READ MORE