Hyperbolic conservation laws on manifolds. Error estimate for finite volume schemes

TL;DR

This paper extends finite volume scheme error analysis for hyperbolic conservation laws from Euclidean spaces to Riemannian manifolds, providing an L1-error estimate that accounts for geometric complexities.

Contribution

It generalizes existing Euclidean error estimates to curved manifolds, incorporating geometric effects into the analysis of finite volume schemes for conservation laws.

Findings

L1-error estimate of order h^(1/4) on manifolds

Extension of Euclidean theory to curved geometries

Overcoming technical challenges due to manifold curvature

Abstract

Following Ben-Artzi and LeFloch, we consider nonlinear hyperbolic conservation laws posed on a Riemannian manifold, and we establish an L1-error estimate for a class of finite volume schemes allowing for the approximation of entropy solutions to the initial value problem. The error in the L1 norm is of order h^(1/4) at most, where h represents the maximal diameter of elements in the family of geodesic triangulations. The proof relies on a suitable generalization of Cockburn, Coquel, and LeFloch's theory which was originally developed in the Euclidian setting. We extent the arguments to curved manifolds, by taking into account the effects to the geometry and overcoming several new technical difficulties.