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A Numerical Study of Diagonally Split Runge–Kutta Methods for PDEs with Discontinuities

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Abstract

Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) time-stepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although second- and third-order unconditionally contractive DSRK methods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like first-order implicit methods. This is unfortunate, because it is precisely to allow a large time-step that we choose to use implicit methods. These results suggest that unconditionally contractive DSRK methods are limited in usefulness as they are unable to compete with either the first-order backward Euler method for large step-sizes or with Crank–Nicolson or high-order explicit SSP Runge–Kutta methods for smaller step-sizes.

We also present stage order conditions for DSRK methods and show that the observed order reduction is associated with the necessarily low stage order of the unconditionally contractive DSRK methods.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Colin B. Macdonald & Steven J. Ruuth

  2. Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, MA, 02747, USA

    Sigal Gottlieb

Authors
  1. Colin B. Macdonald
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  2. Sigal Gottlieb
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  3. Steven J. Ruuth
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Corresponding author

Correspondence to Colin B. Macdonald.

Additional information

The work of C.B. Macdonald was partially supported by an NSERC Canada PGS-D scholarship, a grant from NSERC Canada, and a scholarship from the Pacific Institute for the Mathematical Sciences (PIMS).

The work of S. Gottlieb was supported by AFOSR grant number FA9550-06-1-0255.

The work of S.J. Ruuth was partially supported by a grant from NSERC Canada.

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Macdonald, C.B., Gottlieb, S. & Ruuth, S.J. A Numerical Study of Diagonally Split Runge–Kutta Methods for PDEs with Discontinuities. J Sci Comput 36, 89–112 (2008). https://doi.org/10.1007/s10915-007-9180-6

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  • DOI: https://doi.org/10.1007/s10915-007-9180-6

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