A New Semi-Discrete Finite-Volume Active Flux Method for Hyperbolic Conservation Laws

Abstract: In this work, we introduce a new active flux (AF) method for hyperbolic systems of conservation laws. Following an AF approach recently proposed in [{\sc R. Abgrall}, Commun. Appl. Math. Comput., 5 (2023), pp. 370--402], we consider two different formulations of the studied system (the original conservative formulation and a primitive one containing nonconservative products), and discretize them on overlapping staggered meshes using two different numerical schemes. The novelty of our method is twofold. First, we introduce an original paradigm making use of overlapping finite-volume (FV) meshes over which cell averages of conservative and primitive variables are evolved using semi-discrete FV methods: The nonconservative system is discretized by a path-conservative central-upwind scheme and its solution is used to evaluate very simple numerical fluxes for the discretization of the original conservative system. Second, to ensure the nonlinear stability of the resulting AF method, we design a post-processing, which also guarantees a conservative coupling between the two sets of variables. We test the proposed semi-discrete FV AF method on a number of benchmarks for the one- and two-dimensional Euler equations of gas dynamics.