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Spectral stability of inviscid roll-waves

Abstract : We carry out a systematic analytical and numerical study of spectral stability of dis-continuous roll wave solutions of the inviscid Saint Venant equations, based on a periodic Evans-Lopatinsky 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.
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Mathew Johnson, Pascal Noble, L Rodrigues, Zhao Yang, Kevin Zumbrun. Spectral stability of inviscid roll-waves. Communications in Mathematical Physics, Springer Verlag, 2019, 367 (1), pp.265-316. ⟨10.1007/s00220-018-3277-7⟩. ⟨hal-01870739⟩

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