Error estimate for time-explicit finite volume approximation of strong solutions to systems of conservation laws

TL;DR

This paper provides an error estimate for time-explicit finite volume schemes approximating strong solutions to nonlinear conservation laws, emphasizing entropy conditions and deriving an $h^{1/4}$ convergence rate.

Contribution

It introduces a novel error analysis framework for explicit finite volume methods on unstructured meshes with entropy satisfying fluxes, under a strengthened CFL condition.

Findings

Established a weak-BV estimate for the numerical approximation.

Derived an $h^{1/4}$ error estimate in $L^2$ norm.

Quantified numerical entropy dissipation at mesh interfaces.

Abstract

We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is quantified at each interface of the mesh, which enables to prove a weak--BV estimate for the numerical approximation under a strengthen CFL condition. Then we derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical $h^1/4$ estimate in $L^2$ under this strengthen CFL condition.