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Two Barriers on Strong-Stability-Preserving Time Discretization Methods

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Abstract

Strong-stability-preserving (SSP) time discretization methods are popular and effective algorithms for the simulation of hyperbolic conservation laws having discontinuous or shock-like solutions. They are (nonlinearly) stable with respect to general convex functionals including norms such as the total-variation norm and hence are often referred to as total-variation-diminishing (TVD) methods. For SSP Runge–Kutta (SSPRK) methods with positive coefficients, we present results that fundamentally restrict the achievable CFL coefficient for linear, constant-coefficient problems and the overall order of accuracy for general nonlinear problems. Specifically we show that the maximum CFL coefficient of an s-stage, order-p SSPRK method with positive coefficients is s−p+1 for linear, constant-coefficient problems. We also show that it is not possible to have an s-stage SSPRK method with positive coefficients and order p>4 for general nonlinear problems.

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Authors and Affiliations

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

    Steven J. Ruuth

  2. Department of Computer Science, Dalhousie University, Halifax, Nova Scotia, B3H 1W5, Canada

    Raymond J. Spiteri

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  1. Steven J. Ruuth
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  2. Raymond J. Spiteri
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Ruuth, S.J., Spiteri, R.J. Two Barriers on Strong-Stability-Preserving Time Discretization Methods. Journal of Scientific Computing 17, 211–220 (2002). https://doi.org/10.1023/A:1015156832269

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