Wave patterns for shallow water equations
Authors:
Chiu-Ya Lan and Huey-Er Lin
Journal:
Quart. Appl. Math. 63 (2005), 225-249
MSC (2000):
Primary 76B15, 76H05, 35L65
DOI:
https://doi.org/10.1090/S0033-569X-05-00939-6
Published electronically:
April 12, 2005
MathSciNet review:
2150771
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We consider the time-asymptotic behavior of the system of shallow water equations with one bump in one dimension. Our main interest is in the issue of nonlinear stability and instability of the waves, particularly for the transonic flow. In this paper, the formation of the asymptotic wave patterns is done by combining elementary nonlinear waves, shock and rarefaction waves for the conservation laws, and stationary waves. We also describe the bifurcations of the wave patterns as the end states vary.
Bouchut-Castelnau F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte, A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows, preprint.
Douady- Andreotti S. Douady, B. Andreotti, and A. Daerr, On granular surface flow equations, Eur. Phys. J. B 11 (1999), 131–142.
- J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 1, 89–102. MR 1821555, DOI https://doi.org/10.3934/dcdsb.2001.1.89
- Alexander Kurganov and Doron Levy, Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal. 36 (2002), no. 3, 397–425. MR 1918938, DOI https://doi.org/10.1051/m2an%3A2002019
- Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
- Wen-Ching Lien, Hyperbolic conservation laws with a moving source, Comm. Pure Appl. Math. 52 (1999), no. 9, 1075–1098. MR 1692156, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199909%2952%3A9%3C1075%3A%3AAID-CPA2%3E3.3.CO%3B2-W
- Tai-Ping Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 (1987), no. 11, 2593–2602. MR 913412, DOI https://doi.org/10.1063/1.527751
Bouchut-Castelnau F. Bouchut, A. Mangeney-Castelnau, B. Perthame, and J.-P. Vilotte, A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows, preprint.
Douady- Andreotti S. Douady, B. Andreotti, and A. Daerr, On granular surface flow equations, Eur. Phys. J. B 11 (1999), 131–142.
Gerbeau J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete and continuous dynamical system- series B 1, no.1 (2001).
Kurganov A. Kurganov and D. Levy, Central-upwind schemes for Saint-Venant system, M2AN Math. Model. Numer. Anal. 36 (2002), no. 3, 397–425.
Lax P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, (1973).
Lien W.-C. Lien, Hyperbolic Conservation Laws with a moving source, Communication on Pure and Applied Mathematics, Vol. LII, (1999), 1075–1098.
Liu T.-P. Liu, Nonlinear resonance for quasilinear hyperbolic equation, J. Math. Phys. 28 no. 11 (1987), 2593–2602.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2000):
76B15,
76H05,
35L65
Retrieve articles in all journals
with MSC (2000):
76B15,
76H05,
35L65
Additional Information
Chiu-Ya Lan
Affiliation:
Institute of Mathematics, Academia Sinica, Nankong, Taipei 11529, Taiwan
Email:
cylan@math.nsysu.edu.tw
Huey-Er Lin
Affiliation:
Institute of Mathematics, Academia Sinica, Nankong, Taipei 11529, Taiwan
Email:
helin@math.ntnu.edu.tw
Received by editor(s):
April 20, 2003
Published electronically:
April 12, 2005
Article copyright:
© Copyright 2005
Brown University