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Let's consider two finite difference schemes for PDEs/ODEs. One is conditionally stable, the other is unconditionally stable. People always prefer unconditionally stable ones to conditionally stable ones because for the former one can take large steps. However, one needs to take small steps to get higher accuracy anyway. Is it possible that, for the same continuous equation to solve and under the same step length (in the stable regime for the conditionally stable scheme), a conditionally stable scheme is more accurate than an unconditionally stable one? If so, examples?

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    $\begingroup$ Sure, for a 1D diffusion equation an explicit time integration scheme would have a CFL constraint on the time step, but an implicit scheme would not. But an explicit high-order (in time and space) scheme would be [formally] more accurate (within its range of stability) than an implicit low-order scheme. $\endgroup$ Jan 25, 2023 at 15:47

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This is absolutely possible. For example, say you're solving the advection equation $$\frac{\partial q}{\partial t} + \nabla\cdot q\,\mathbf u = 0$$ for the field $q$ using your favorite technique for spatial discretization. One of the 3rd-order strong stability-preserving Runge-Kutta methods is going to be more accurate than, say, a 1st-order implicit scheme despite the fact that you can take longer timesteps with the implicit scheme. On top of that, the fact that you can take longer timesteps with an implicit method doesn't necessarily mean that you'll get a very good answer, it just means that you won't get an array of NaNs.

But I think there's also more to the story than just formal order of accuracy. For starters, with some pretty basic conditions on the spatial discretization, the 1st-order implicit scheme has a maximum principle -- the solution will not develop new local maxima and minima wherever the true solution does not. This is especially important when the true solution has strict bounds that are required for well-posedness of other parts of the dynamics, for example the thickness of a fluid film remaining positive. Many commonly-used higher-order schemes don't have maximum principles.

There are also human factors to consider. Are you implementing numerical methods for your own use, and if so do you really want to spend time tweaking the timesteps of every simulation you run? Or are you going to hand off this code to a physical scientist who has never even heard of the Courant-Friedrichs-Lewy condition? Is the gain in speed worth the loss in reliability and robustness?

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  • $\begingroup$ thanks a lot for the great answer. Are all unconditionally stable schemes dissipative and if so won't be good for energy preserving equations? At least conditionally stable schemes offer an option to tune energy preserving, so could be better in this respect? $\endgroup$
    – feynman
    Jan 26, 2023 at 3:27
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    $\begingroup$ Ok, what about the implicit midpoint method $(z_{n + 1} - z_n) / \delta t = f((z_{n + 1} + z_n) / 2)$? What is its stability region? Is it dissipative? Is it symplectic? $\endgroup$ Jan 26, 2023 at 4:36
  • $\begingroup$ thanks, what's the conclusion for this implicit midpoint method? $\endgroup$
    – feynman
    Jan 28, 2023 at 4:53

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