Large time step TVD schemes for hyperbolic conservation laws
For researchers in numerical methods for hyperbolic PDEs, this work provides a theoretical characterization and practical construction of large time step TVD schemes, though it is incremental as it extends existing frameworks.
The paper proposes a one-parameter family of large time step TVD schemes for hyperbolic conservation laws, spanning the full range of admissible total numerical viscosity, and proves that all local unconditionally TVD schemes have numerical viscosity coefficients bounded by those of LTS-Roe and LTS-LxF schemes. Numerical tests on Burgers and Euler equations demonstrate the framework.
Large time step explicit schemes in the form originally proposed by LeVeque 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 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 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 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 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.