Two Barriers on Strong-Stability-Preserving Time Discretization Methods

  • Authors:

  • Steven J. Ruuth;Raymond J. Spiteri

  • Affiliations:

  • Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6 Canada. sruuth@sfu.ca;Department of Computer Science, Dalhousie University, Halifax, Nova Scotia, B3H 1W5 Canada. spiteri@cs.dal.ca

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2002

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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 p4 for general nonlinear problems.