Spectral Stability of Inviscid Roll Waves

TL;DR

This paper provides a comprehensive analysis of the spectral stability of inviscid roll wave solutions in the Saint Venant equations, combining analytical and numerical methods to produce a detailed stability diagram relevant to hydraulic engineering.

Contribution

It introduces a novel spectral stability analysis using a periodic Evans-Lopatinski determinant and establishes explicit stability boundaries that align with viscous case results.

Findings

Explicit low-frequency stability boundary derived

Stability boundary closely matches viscous case predictions

Spectral stability diagram applicable in hydraulic engineering

Abstract

We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint Venant equations, based on a periodic Evans-Lopatinski determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but not implied by the associated formal first-order Whitham modulation equations.