High Order Strong Stability Preserving Time Discretizations

  • Authors:

  • Sigal Gottlieb;David I. Ketcheson;Chi-Wang Shu

  • Affiliations:

  • Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth, USA 02747;Department of Applied Mathematics, University of Washington, Seattle, USA 98195-2420;Division of Applied Mathematics, Brown University, Providence, USA 02912

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2009

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Abstract

Strong stability preserving (SSP) high order time discretizations were developed to ensure nonlinear stability properties necessary in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. SSP methods preserve the strong stability properties--in any norm, seminorm or convex functional--of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the connections between the timestep restrictions for strong stability preservation and contractivity. Numerical examples demonstrate that common linearly stable but not strong stability preserving time discretizations may lead to violation of important boundedness properties, whereas SSP methods guarantee the desired properties provided only that these properties are satisfied with forward Euler timestepping. We review optimal explicit and implicit SSP Runge---Kutta and multistep methods, for linear and nonlinear problems. We also discuss the SSP properties of spectral deferred correction methods.