Spurious caustics of Dispersion Relation Preserving schemes

TL;DR

This paper investigates spurious caustics in dispersion-relation preserving schemes, revealing how they cause error bursts in numerical simulations due to extrema in numerical group velocity, extending previous work on classical schemes.

Contribution

It identifies and analyzes spurious caustics in dispersion-relation preserving schemes, linking them to extrema in numerical group velocity and extending prior findings to these schemes.

Findings

Spurious caustics cause error bursts in numerical simulations.

Extrema of numerical group velocity are linked to spurious caustics.

The study extends previous work to dispersion-relation preserving schemes.

Abstract

A linear dispersive mechanism leading to a burst in the $L_\infty$ norm of the error in numerical simulation of polychromatic solutions is identified. This local error pile-up corresponds to the existence of spurious caustics, which are allowed by the dispersive nature of the numerical error. From the mathematical point of view, spurious caustics are related to extrema of the numerical group velocity and are physically associated to interactions between rays defined by the characteristic lines of the discrete system. This paper extends our previous work about classical schemes to dispersion-relation preserving schemes.