Conditional a priori error estimates of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D

TL;DR

This paper establishes conditional error estimates for finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting applied to 1D hyperbolic conservation laws, under specific regularity and stability conditions.

Contribution

It provides the first conditional a priori error estimates for these methods with abstract limiting, linking convergence rates to solution regularity and shock trace accuracy.

Findings

Convergence in L-infinity L-one norm with rate h^{1/3}

Error estimates depend on shock trace accuracy and solution regularity

Method achieves convergence under specified mesh and solution conditions

Abstract

We derive conditional a priori error estimates of a wide class of finite volume and Runge-Kutta discontinuous Galerkin methods with abstract limiting for hyperbolic systems of conservation laws in 1D via the verification of weak consistency and entropy stability, as recently proposed by Bressan et al.~\cite{BressanChiriShen21}. Convergence in $L^\infty L^1$ with rate $h^{1/3}$ is obtained under a time step restriction $\tau\leq ch$, provided the following conditions hold: the exact solution is piecewise Lipschitz continuous, its (finitely many and isolated) shock curves can be traced with precision $h^{2/3}$ and, outside of these shock tracing tubular neighborhoods the numerical solution -- assumed to be uniformly small in BV -- has oscillation strength $h$ across each mesh cell and cell boundary.