Optimal Strong-Stability-Preserving Time-Stepping Schemes with Fast Downwind Spatial Discretizations

In the field of strong-stability-preserving time discretizations, a number of researchers have considered using both upwind and downwind approximations for the same derivative, in order to guarantee that some strong stability condition will be preserved. The cost of computing both the upwind and downwind operator has always been assumed to be double that of computing only one of the two. However, in this paper we show that for the weighted essentially non-oscillatory method it is often possible to compute both these operators at a cost that is far below twice the cost of computing only one. This gives rise to the need for optimal strong-stabilitypreserving time-stepping schemes which take into account the different possible cost increments. We construct explicit linear multistep schemes up to order six and explicit Runge-Kutta schemes up to order four which are optimal over a range of incremental costs.