Large Time Step TVD Schemes for Hyperbolic Conservation Laws

Abstract

Large time step (LTS) explicit schemes in the form originally proposed by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463--477] have seen a significant revival in recent years. In this paper we consider a general framework of local (2k+1)-point schemes containing LeVeque's scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient. We identify the least and most diffusive total variation diminishing (TVD) schemes in this framework. The most diffusive scheme is the(2k+1)-point Lax--Friedrichs scheme (LTS-LxF). The least diffusive scheme is the LTS scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes. We discuss the nature of entropy violations associated with the LTS-Roe scheme; in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions and discuss numerical entropy fixes. Finally, we propose a one-parameter family of LTS TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The one-dimensional Burgers equation and the Euler system are used as numerical illustrations.