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

Sigal Gottlieb, Zachary Grant, Daniel Higgs

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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.
Original languageEnglish (US)
Pages (from-to)2743-2761
Number of pages19
JournalMathematics of Computation
Issue number296
StatePublished - Apr 10 2015
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): FIC/2010/05
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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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