# Adaptive Timestepping for Stong Stability Preserving (SSP) Runge-Kutta Methods

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?

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.