Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary

TL;DR

This paper proves the convergence of the finite volume method for nonlinear hyperbolic conservation laws on spacetimes with boundary, extending previous work by including boundary slices and introducing total flux functions.

Contribution

It introduces a new finite volume formulation with total flux functions and proves convergence and well-posedness for conservation laws on bounded spacetimes.

Findings

Finite volume method converges for weak solutions with entropy conditions.

Existence and uniqueness of solutions are established under hyperbolicity and foliation assumptions.

A contraction property in an L1-type distance is demonstrated.

Abstract

We study nonlinear hyperbolic conservation laws posed on a differential (n+1)-manifold with boundary referred to as a spacetime, and defined from a prescribed flux field of n-forms depending on a parameter (the unknown variable), a class of equations proposed by LeFloch and Okutmustur in 2008. Our main result is a proof of the convergence of the finite volume method for weak solutions satisfying suitable entropy inequalities. A main difference with previous work is that we allow for slices with a boundary and, in addition, introduce a new formulation of the finite volume method involving the notion of total flux functions. Under a natural global hyperbolicity condition on the flux field and the spacetime and by assuming that the spacetime admits a foliation by compact slices with boundary, we establish an existence and uniqueness theory for the initial and boundary value problem, and we prove a contraction property in a geometrically natural L1-type distance.