Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order

Type
Article

Authors
Gottlieb, Sigal
Grant, Zachary
Higgs, Daniel

KAUST Grant Number
FIC/2010/05

Date
2015-04-10

Abstract
High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge-Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge-Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge-Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may still be worthwhile to use methods with higher nonlinear order.

Citation
Gottlieb S, Grant Z, Higgs D (2015) Optimal explicit strong stability preserving Runge–Kutta methods with high linear order and optimal nonlinear order. Math Comp 84: 2743–2761. Available: http://dx.doi.org/10.1090/mcom/2966.

Acknowledgements
The authors wish to thank Professor Bram van Leer for the motivation for studying this problem, and Professor David Ketcheson for many helpful discussions. This publication is based on work supported by AFOSR grant FA-9550-12-1-0224 and KAUST grant FIC/2010/05.

Publisher
American Mathematical Society (AMS)

Journal
Mathematics of Computation

DOI
10.1090/mcom/2966

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