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Many numerical tips and theoretical explanations can be found in this book from Hairer and Wanner: https://www.springer.com/gp/book/9783540566700 In this book, a strategy is described, which uses a time step such that the relative variation of the solution during the first time step is below a certain threshold if you were using explicit Euler (omitting the ...


1

As stated there's no re-rejection mechanism, i.e. ability to decrease the stepsize after a step has potentially failed. This is required for implicit methods which have Newton steps since there's a chance the $\Delta t$ is large enough that the (quasi-)Newton is unstable, in which case it needs to pullback on time. This instability can sometimes be seen via ...


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I do not think this is a bad approach, but it is not a very precise way to select timesteps either. Admittedly, I have not come across this sort of timestep heuristic before, but looking at a linear test problem provides some insight as to why this is reasonable. For $y' = \lambda y$, the conditions becomes $\Delta t = \frac{\alpha}{\lambda}$. This looks ...


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