Spectral Stability of Inviscid Roll Waves

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.

arXiv comment45 pages, 11 figures

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• Validity of Whitham's modulation equations for dissipative systems with a conservation law -- Phase dynamics in a generalized Ginzburg-Landau system --
• Majda and ZND models for detonation: nonlinear stability vs. formation of singularities