An Innovative Finite Volume Method for Discretizing Hyperbolic Systems of Conservations Laws

Abstract

This work describes a novel subface flux-based Finite Volume (FV) method for discretizing multi- dimensional hyperbolic systems of conservation laws over general unstructured grids. The subface flux numerical approximation relies on the notion of simple Eulerian Riemann solver introduced in the sem- inal work [2]. The Eulerian Riemann solver is constructed from its Lagrangian counterpart by means of the Lagrange-to-Euler mapping. This systematic procedure ensures the transfer of good properties such as positivity preservation and entropy stability [2, 1]. In this framework, the conservativity and the entropy stability are no more locally face-based but result respectively from a node-based vectorial equation and a scalar inequation. The corresponding multi-dimensional FV scheme is characterized by an explicit time step condition ensuring positivity preservation and entropy stability. The application to gas dynamics provides an original multi-dimensional conservative and entropy-stable FV scheme wherein the numerical fluxes are computed through a nodal solver which is similar to the one designed for Lagrangian hydrodynamics [3]. We also observe that the present approach relies on an approximate Riemann solver for Eulerian gas dynamics characterized by naturally ordered wave speeds contrarily to the one introduced in [4]. The robustness and the accuracy of this novel FV scheme are assessed through various numerical tests. We observe its insensitivity to the numerical pathologies that plague classical face-based contact discontinuity preserving FV formulations.