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Journal of Scientific Computing

, Volume 38, Issue 3, pp 251–289 | Cite as

High Order Strong Stability Preserving Time Discretizations

  • Sigal Gottlieb
  • David I. Ketcheson
  • Chi-Wang Shu
Review Article

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.

Keywords

Strong stability preserving Runge–Kutta methods Multistep methods Spectral deferred correction methods High order accuracy Time discretization 

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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sigal Gottlieb
    • 1
  • David I. Ketcheson
    • 2
  • Chi-Wang Shu
    • 3
  1. 1.Department of MathematicsUniversity of Massachusetts DartmouthNorth DartmouthUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA
  3. 3.Division of Applied MathematicsBrown UniversityProvidenceUSA

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