Coupling techniques for nonlinear hyperbolic equations. IV. Multi-component coupling and multidimensional well-balanced schemes

TL;DR

This paper develops a multidimensional, multi-component coupling framework for nonlinear hyperbolic equations, introducing well-balanced finite volume schemes and proving their convergence to entropy solutions.

Contribution

It extends previous coupling methods to multiple space dimensions and components, and introduces a well-balanced finite volume scheme with convergence analysis.

Findings

Designed a well-balanced finite volume method for coupled hyperbolic equations.

Proved convergence of the scheme to entropy solutions in multiple dimensions.

Extended coupling theory to cover multiple conservation laws and space coverings.

Abstract

This series of papers is devoted to the formulation and the approximation of coupling problems for nonlinear hyperbolic equations. The coupling across an interface in the physical space is formulated in term of an augmented system of partial differential equations. In an earlier work, this strategy allowed us to develop a regularization method based on a thick interface model in one space variable. In the present paper, we significantly extend this framework and, in addition, encompass equations in several space variables. This new formulation includes the coupling of several distinct conservation laws and allows for a possible covering in space. Our main contributions are, on one hand, the design and analysis of a well-balanced finite volume method on general triangulations and, on the other hand, a proof of convergence of this method toward entropy solutions, extending Coquel, Cockburn, and LeFloch's theory (restricted to a single conservation law without coupling). The core of our analysis is, first, the derivation of entropy inequalities as well as a discrete entropy dissipation estimate and, second, a proof of convergence toward the entropy solution of the coupling problem.