Optimal implicit strong stability preserving Runge-Kutta methods

David I. Ketcheson*, Colin B. Macdonald, Sigal Gottlieb

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

82 Scopus citations


Strong stability preserving (SSP) time discretizations were developed for use with spatial discretizations of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified timestep restriction. We turn to implicit Runge-Kutta methods to alleviate this timestep restriction, and present implicit SSP Runge-Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of SSP Runge-Kutta methods) and up to eleven stages. The numerically optimal methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge-Kutta methods are diagonally implicit. These methods allow a larger SSP timestep, compared to explicit methods of the same order and number of stages. Numerical tests verify the order and the SSP property of the methods.

Original languageEnglish (US)
Pages (from-to)373-392
Number of pages20
JournalApplied Numerical Mathematics
Issue number2
StatePublished - Feb 2009
Externally publishedYes


  • Runge-Kutta methods
  • Strong stability preserving
  • Time discretization

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


Dive into the research topics of 'Optimal implicit strong stability preserving Runge-Kutta methods'. Together they form a unique fingerprint.

Cite this