Global optimization of explicit strong-stability-preserving Runge-Kutta methods

Abstract

Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially 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., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to fifth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coefficients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (specifically, for fifth-order methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature.