Total variation diminishing Runge-Kutta schemes

1998/01/01 by Sigal Gottlieb, Chi-Wang Shu, Chi‐Wang Shu · 2,107 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Computational Fluid Dynamics and Aerodynamics #Conservation law #Discretization #Fluid Dynamics and Turbulent Flows #Mathematical analysis #Mathematics #Numerical analysis #Physics #Runge–Kutta methods #Total variation diminishing #Variation (astronomy)

paper · pdf · doi:10.1090/s0025-5718-98-00913-2

published in Mathematics of Computation 67(221), 73-85 (American Mathematical Society (AMS))

crossref issued 1998/01/01 · crossref published 1998/01/01 · crossref published-online 1998/01/01 · openalex publication_date 1998/01/01 · crossref created 2002/07/26 · openalex created_date 2025/10/10 · crossref deposited 2026/04/20 · openalex updated_date 2026/08/02 · crossref indexed 2026/08/07

Abstract

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

Citations

• Efficient implementation of essentially non-oscillatory shock-capturing schemes

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