On the Strong Stability Preserving Property of Runge-Kutta Methods for Hyperbolic Problems

TL;DR

This paper develops a mathematical framework to analyze the nonlinear stability of Runge-Kutta schemes, revealing that many schemes outside the SSP class can still preserve key stability properties for hyperbolic problems.

Contribution

It introduces a new approach to assess the stability of non-SSP Runge-Kutta methods, expanding the class of schemes suitable for hyperbolic PDEs.

Findings

Classical RK schemes can maintain entropy stability and positivity.

Broader RK methods can ensure TVD stability for certain schemes.

Numerical experiments validate the theoretical stability analysis.

Abstract

Strong Stability Preserving (SSP) time integration schemes maintain stability of the forward Euler method for any initial value problem. However, only a small subset of Runge-Kutta (RK) methods are SSP, and many efficient high-order time integration schemes do not formally belong to this class. In this work, we introduce a mathematical strategy to analyze the nonlinear stability of RK schemes that may not necessarily belong to the SSP class. With this approach, we mathematically demonstrate that there are time integration schemes outside the class of SSP schemes that can maintain entropy stability and positivity of density and pressure for the Lax-Friedrichs discretization, and Total Variation Diminishing stability for the first-order upwind and the second-order MUSCL schemes. As a result, for these problems, a broader range of RK methods, including the classical fourth-order, four-stage RK scheme, can be used while the numerical integration remains stable. Numerical experiments confirm these theoretical findings, and additional experiments demonstrate similar observations for a wider class of space discretizatins.