A posteriori Error Estimates for Numerical Solutions to Hyperbolic Conservation Laws

TL;DR

This paper develops a posteriori error estimates for numerical solutions to hyperbolic conservation laws, using a post-processing algorithm that assesses total variation and oscillation, applicable to various schemes like Godunov and Lax-Friedrichs.

Contribution

It introduces a new post-processing method for reliable a posteriori error bounds applicable to multiple numerical schemes for hyperbolic laws.

Findings

The estimates effectively bound errors in numerical solutions.

The method applies to schemes like Godunov, Lax-Friedrichs, and backward Euler.

Numerical examples demonstrate the approach's practicality.

Abstract

The paper is concerned with a posteriori error bounds for a wide class of numerical schemes, for $n\times n$ hyperbolic conservation laws in one space dimension. These estimates are achieved by a "post-processing algorithm", checking that the numerical solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, in particular, to solutions obtained by the Godunov or the Lax-Friedrichs scheme, backward Euler approximations, and the method of periodic smoothing. Some numerical implementations are presented.