Dual Formulation Finite-Volume Methods on Overlapping Meshes for Hyperbolic Conservation Laws

TL;DR

This paper presents innovative second-order finite-volume schemes for hyperbolic conservation laws using overlapping meshes and dual formulations, enhancing stability and accuracy in gas dynamics simulations.

Contribution

It introduces a novel overlapping mesh approach with dual formulations and a post-processing step to ensure stability and conservation.

Findings

Effective on Euler gas dynamics benchmarks

Achieves nonlinear stability through post-processing

Utilizes simple flux evaluations for efficiency

Abstract

In this work, we introduce new second-order schemes for one- and two-dimensional hyperbolic systems of conservation laws. Following an 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 approach 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 method, we design a post-processing, which also guarantees a conservative coupling between the two sets of variables. We test the proposed semi-discrete dual formulation finite-volume methods on several benchmarks for the Euler equations of gas dynamics.