# Strong-Stability preserving Runge-Kutta time-steppers¶

The Gkyl DG solvers use SSP-RK time-steppers. Three steppers are implemented: SSP-RK2, SSP-RK3 and a four-stage SSP-RK3 that allows twice the CFL (for the cost of additional memory) as the other schemes. See [DurranBook] page 56. The schemes are described below. Here, the symbol \(\mathcal{F}\) is used to indicate a first-order Euler update:

where \(\mathcal{L}[f]\) is the RHS operator from the spatial discretization of the DG scheme.

Contents

## SSP-RK2¶

with \(CFL \le 1\).

## SSP-RK3¶

with \(CFL \le 1\). As this scheme has three stages instead of two, it will take about \(1.5X\) longer to run than the SSP-RK2 scheme.

## Four stage SSP-RK3¶

with \(CFL\le 2\). Note that this scheme has four stages, but allows twice the time-step that SSP-RK2 and SSP-RK3, hence will result in a speed up of \(1.5X\) compared to the three-stage SSP-RK3 scheme.

## Region of absolute stability¶

For each of the above schemes, I have plotted below the region of absolute stability. Note that only the RK3 schemes are stable when there is no diffusion in the system, and hence should be prefered.

## References¶

[DurranBook] | Dale E. Durran, “Numerical Methods for Fluid Dynamics”, Springer. Second Edition. |