A unified Riemann-problem-based extension of the Warming-Beam and Lax-Wendroff schemes (Q2785707)

The authors present a TVD Riemann-problem-based numerical method for solving systems of hyperbolic conservation laws in one space dimension. It is a single Godunov-type method based on a generalization of the weighted average flux obtained via a space-time integral of solutions of local Riemann problems. In fact, the scheme is an extension of the method of \textit{R. F. Warming} and \textit{R. M. Beam} [AIAA J. 14, 1241-1249 (1976; Zbl 0364.76047)] if the CFL number lies between 1 and 2, and of the Lax-Wendroff method if it lies between 0 and 1. The local wave structure dictates switching between schemes automatically with no need for special conservative switching operator. The method is one-dimensional, but can be extended to the multi-dimensional scheme via space operator splitting. Numerical experiments confirm reliability and robustness of the proposed scheme.