Strong Stability-Preserving High-Order Time Discretization Methods

2001/01/01 by Sigal Gottlieb, Chi-Wang Shu, Chi‐Wang Shu +1 · 2,545 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Backward Euler method #Class (philosophy) #Computational Fluid Dynamics and Aerodynamics #Computer science #Differential algebraic equation #Differential equation #Discretization #Euler equations #Euler method #Euler's formula #Hyperbolic partial differential equation #Linear multistep method #Mathematical analysis #Mathematics #Nonlinear system #Numerical methods for differential equations #Ordinary differential equation #Partial differential equation #Physics #Runge–Kutta methods #Stability (learning theory) #Total variation diminishing

paper · doi:10.1137/s003614450036757x

published in SIAM Review 43(1), 89-112 (Society for Industrial and Applied Mathematics)

openalex publication_date 2001/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

Citations

• Total variation diminishing Runge-Kutta schemes
• Efficient implementation of essentially non-oscillatory shock-capturing schemes

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