Richardson Extrapolation for Linearly Degenerate Discontinuities

TL;DR

This paper explores the effectiveness of Richardson extrapolation in estimating convergence rates for numerical advection solutions with discontinuities, identifying specific cases where it provides accurate results through analysis and numerical demonstrations.

Contribution

The paper analyzes Richardson extrapolation's applicability to discontinuous advection problems and identifies conditions under which it yields correct convergence estimates.

Findings

Richardson extrapolation often does not match a-priori convergence estimates.

A specific use case is identified where Richardson extrapolation accurately estimates convergence rates.

Numerical examples confirm the theoretical findings.

Abstract

In this paper we investigate the use of Richardson extrapolation to estimate the convergence rates for numerical solutions to advection problems involving discontinuities. We use modified equation analysis to describe the expectation of the approach. In general, the results do not agree with a-priori estimates of the convergence rates. However, we identify one particular use case where Richardson extrapolation does yield the proper result. We then demonstrate this result using a number of numerical examples.