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Communications in Mathematical Physics

, Volume 367, Issue 1, pp 265–316 | Cite as

Spectral Stability of Inviscid Roll Waves

  • Mathew A. JohnsonEmail author
  • Pascal Noble
  • L. Miguel Rodrigues
  • Zhao Yang
  • Kevin Zumbrun
Article

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–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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Notes

Acknowledgements

Thanks to Olivier Lafitte for stimulating discussions regarding normal forms for singular ODE, and to Blake Barker for his generous help in sharing source computations from [BJN+17]. The numerical computations in this paperwere carried out in the MATLAB environment; analytical calculations were checked with the aid of MATLAB’s symbolic processor. Thanks to University Information Technology Services (UITS) division from Indiana University for providing the Karst supercomputer environment in which most of our computations were carried out. This research was supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute, and in part by the Indiana METACyt Initiative. The Indiana METACyt Initiative at IU was also supported in part by Lilly Endowment, Inc.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of KansasLawrenceUSA
  2. 2.Institut deMathématiques de ToulouseUMR5219, CNRS, INSA, Université de ToulouseToulouseFrance
  3. 3.Univ Rennes, CNRS, IRMAR - UMR 6625RennesFrance
  4. 4.Indiana UniversityBloomingtonUSA

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