Improving the quantification of overshooting shock-capturing oscillations

TL;DR

This paper introduces a quantitative method to evaluate overshooting oscillations in shock-capturing schemes, accounting for both discontinuities and smooth waves, and identifies the wavenumber ranges where schemes are most prone to overshoot.

Contribution

It extends previous work by providing a standardized measurement for overshooting oscillations in mixed discontinuity and smooth wave scenarios, improving robustness analysis.

Findings

Identifies wavenumber ranges prone to overshoot.

Provides a simple measurement for shock-capturing robustness.

Analyzes weighted essentially non-oscillatory schemes.

Abstract

An approach for quantitatively evaluating overshooting oscillations is designed to characterize the performance of shock-capturing schemes. Specifically, following our previous work focused on cases with only discontinuities, now we account for the concurrent presences of discontinuities and smooth waves, each with a complete set of supported modes on a given discretization. The linear advection equation is taken as the model equation, and a standardized measurement is given for overshooting oscillations produced by shock-capturing schemes. Thereby, we can quantitatively reveal the shock-capturing robustness of, for example, weighted essentially non-oscillatory schemes, by comparing and analyzing the resulting overshoots. In particular, we are able to find out the ranges of wavenumbers in which the numerical schemes are especially prone to produce overshooting oscillations. While lower dissipation is usually anticipated for high-order schemes, we provide a simple measurement for evaluating the shock-capturing robustness, which was not trivial due to the nonlinearity of shock-capturing computations.