Large time step explicit schemes in the form originally proposed by LeVeque (Comm. Pure Appl. Math., 37 (1984), pp. 463{477) have seen a signicant 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 modied equation analysis allows us to interpret each numerical cell interface coecient of the framework as a partial numerical viscosity coecient. We identify the least and most diusive TVD schemes in this framework. The most diusive scheme is the 2k + 1-point Lax-Friedrichs scheme (LTS-LxF). The least diusive scheme is the Large Time Step scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coecients of any local unconditionally TVD scheme are bounded by the values of the corresponding coecients of the LTS-Roe and LTS-LxF schemes. We discuss the nature of entropy violations associated with the LTS-Roe scheme, in partic- ular 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 gener- alized transonic rarefactions, and discuss numerical entropy xes. Finally, we propose a one-parameter family of Large Time Step TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The 1D Burgers equation and the Euler system are used as numerical illustrations.
['Sofia Lindqvist', 'Peder Aursand', 'Tore Flåtten', 'Anders Aase Solberg']
Large time step TVD schemes for hyperbolic conservation laws