High-order linear multistep methods with general monotonicity and boundedness properties

  • Authors:

  • Steven J. Ruuth;Willem Hundsdorfer

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6;CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2005

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Abstract

We consider linear multistep methods that possess general monotonicity and boundedness properties. Strict monotonicity, in terms of arbitrary starting values for the multistep schemes, is only valid for a small class of methods, under very stringent step size restrictions. This makes them uncompetitive with the strong-stability-preserving (SSP) Runge-Kutta methods. By relaxing these strict monotonicity requirements a larger class of methods can be considered, including many methods of practical interest. In this paper we construct linear multistep methods of high-order (up to six) that possess relaxed monotonicity or boundedness properties with optimal step size conditions. Numerical experiments show that the new schemes perform much better than the classical monotonicity-preserving multistep schemes. Moreover there is a substantial gain in efficiency compared to recently constructed SSP Runge-Kutta (SSPRK) methods.