Are there error estimators and research on adaptive timestepping schemes for SSPRK methods? My Googling could not uncover papers which addressed this, so I was wondering if there was anything particularly difficult about deriving such error estimators. I think the difficult thing is that the SSP property only occurs when $$\Delta t < c \Delta t_{FE}$$ (in @DavidKetcheson's notation), and so part of the adaptivity would likely need to ensure it stays in this region. Are there any published attempts to do this? And any research on the effects of PI-controllers in this case?
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asked Jun 19, 2017 at 4:21
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SSP methods are mainly used for integrating ODEs corresponding to nonlinear hyperbolic PDE semi-discretizations. In such ODEs, $\Delta t_{FE}$ depends on the solution and so it varies at each time step. In all implementations I know of, SSPRK time stepping is done adaptively in order to ensure that the SSP time step constraint (stated in the question above) is satisfied. This isn't difficult since usually $\Delta t_{FE}$ is just inversely proportional to the maximum characteristic speed.
Some of the words in your question suggest that you are also interested in adapting the timestep to satisfy a local error tolerance. This is not usually done, since for hyperbolic problems with shocks the spatial error typically dominates the temporal error. Nevertheless, having some bound on the temporal error could certainly be useful, and there might be situations in which the error control restricts the time step more tightly than the SSP condition.
A recent preprint that contains several embedded SSP pairs for this purpose is available from arXiv.
I'll also mention that for SSP linear multistep methods, adapting the step size while maintaining the SSP condition is quite challenging; see this paper.
answered Jun 22, 2017 at 4:43