High-Order Strong-Stability-Preserving Runge--Kutta Methods with Downwind-Biased Spatial Discretizations

Abstract

Strong-stability-preserving Runge--Kutta (SSPRK) methods are a specific type of time discretization method that have been widely used for the time evolution of hyperbolic partial differential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE, e.g., stability with respect to total variation, the maximum norm, or other convex functionals. This is of particular interest when the solution exhibits shock-like or other nonsmooth behavior. Many results are known for SSPRK methods with nonnegative coefficients. However, it has recently been shown that such methods cannot exist with order greater than 4. In this paper, we give a systematic treatment of explicit SSPRK methods with general (i.e., possibly negative) coefficients up to order 5. In particular, we show how to optimally treat negative coefficients (corresponding to a change in the upwind direction of the spatial discretization) in the context of effective CFL coefficient maximization and provide proofs of optimality of some explicit SSPRK methods of orders 1 to 4. We also give the first known explicit fifth-order SSPRK schemes and show their effectiveness in practice versus more well-known fifth-order explicit Runge--Kutta schemes.