Exact Riemann solutions to shallow water equations
Authors:
Ee Han and Gerald Warnecke
Journal:
Quart. Appl. Math. 72 (2014), 407-453
MSC (2010):
Primary 76B15, 76H05, 35L60
DOI:
https://doi.org/10.1090/S0033-569X-2014-01353-3
Published electronically:
June 13, 2014
MathSciNet review:
3237558
Full-text PDF Free Access
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Abstract: We determine completely the Riemann solutions to the shallow water equations with a bottom step, including the dry bed problem. The nonstrict hyperbolicity of this first-order system of partial differential equations leads to resonant waves and nonunique solutions. To address these difficulties we construct the L–M and R–M curves in the state space. For the bottom step elevated from left to right, we classify the L–M curve into five different cases and the R–M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the L–M and R–M curves are fully analyzed. We observe that the non–uniqueness of the Riemann solutions is due to bifurcations on the L–M or R–M curves. The possible solutions including classical waves and resonant waves as well as dry bed state are solved in a uniform framework for any given Riemann initial data.
References
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References
- Ashwin Chinnayya, Alain-Yves LeRoux, and Nicolas Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: The resonance phenomenon, Int. J. Finite Vol. 1 (2004), no. 1, 33. MR 2465450 (2009j:65186)
- J.J. Stoker, Water Waves, Interscience,. New York, 1957.
- Nikolai Andrianov, Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations, Internat. J. Numer. Methods Fluids 47 (2005), no. 8-9, 825–831. MR 2123795 (2005i:76079), DOI https://doi.org/10.1002/fld.846
- Nikolai Andrianov and Gerald Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model, J. Comput. Phys. 195 (2004), no. 2, 434–464. MR 2046106 (2004m:76165), DOI https://doi.org/10.1016/j.jcp.2003.10.006
- Francisco Alcrudo and Fayssal Benkhaldoun, Exact solutions to the Riemann problem of the shallow water equations with a bottom step, Comput. & Fluids 30 (2001), no. 6, 643–671. MR 1859269 (2002g:76012), DOI https://doi.org/10.1016/S0045-7930%2801%2900013-5
- F. Benkhaldoun and L. Quivy, A Non Homogeneous Riemann Solver for shallow water and two phase flows, Flow, Turbulence and Combustion 76 (2006), pp. 391-402.
- R. Bernetti, V. A. Titarev, and E. F. Toro, Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry, J. Comput. Phys. 227 (2008), no. 6, 3212–3243. MR 2392731 (2009a:76015), DOI https://doi.org/10.1016/j.jcp.2007.11.033
- R. W. D. Nickalls, A New Approach to Solving the Cubic: Cardan’s Solution Revealed, The Mathematical Gazette, 77 (Nov., 1993), pp. 354-359.
- E. Han, M. Hantke, and G. Warnecke, Exact Riemann solutions in ducts with discontinuous cross–section, J. Hyp. Diff. Equations 9 (2012), pp. 403-449.
- Jiequan Li and Guoxian Chen, The generalized Riemann problem method for the shallow water equations with bottom topography, Internat. J. Numer. Methods Engrg. 65 (2006), no. 6, 834–862. MR 2200026 (2006j:76022), DOI https://doi.org/10.1002/nme.1471
- E. F. Toro, The Dry–Bed Problem in Shallow–Water Flows, College of Aeronautics Reports, 1990.
- E. F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley and Sons. Ltd., 2001.
- Eli Isaacson and Blake Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 52 (1992), no. 5, 1260–1278. MR 1182123 (93f:35140), DOI https://doi.org/10.1137/0152073
- Jihwan Kim and Randall J. LeVeque, Two-layer shallow water system and its applications, Hyperbolic problems: Theory, numerics and applications, Proc. Sympos. Appl. Math., vol. 67, Amer. Math. Soc., Providence, RI, 2009, pp. 737–743. MR 2605269 (2011e:76024)
- Paola Goatin and Philippe G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 6, 881–902 (English, with English and French summaries). MR 2097035 (2006i:35225), DOI https://doi.org/10.1016/j.anihpc.2004.02.002
- V. V. Ostapenko, Dam-break flows over a bottom step, Prikl. Mekh. Tekhn. Fiz. 44 (2003), no. 4, 51–63 (Russian, with Russian summary); English transl., J. Appl. Mech. Tech. Phys. 44 (2003), no. 4, 495–505. MR 2009465 (2004e:76026), DOI https://doi.org/10.1023/A%3A1024292822989
- D. Rochette, S. Clain, and W. Bussière, Unsteady compressible flow in ducts with varying cross–section: Comparison between the nonconservative Euler system and axisymmetric flow model. Preprint submitted.
- Philippe G. LeFloch, Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues, Nonlinear partial differential equations and hyperbolic wave phenomena, Contemp. Math., vol. 526, Amer. Math. Soc., Providence, RI, 2010, pp. 237–272. MR 2731995 (2012d:35222), DOI https://doi.org/10.1090/conm/526/10384
- Philippe G. LeFloch and Mai Duc Thanh, The Riemann problem for the shallow water equations with discontinuous topography, Commun. Math. Sci. 5 (2007), no. 4, 865–885. MR 2375051 (2009k:35179)
- Philippe G. LeFloch and Mai Duc Thanh, A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J. Comput. Phys. 230 (2011), no. 20, 7631–7660. MR 2823568 (2012g:76028), DOI https://doi.org/10.1016/j.jcp.2011.06.017
- D. Marchesin and P. J. Paes-Leme, A Riemann problem in gas dynamics with bifurcation: Hyperbolic partial differential equations, III, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 433–455. MR 841979 (87e:76106)
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Additional Information
Ee Han
Affiliation:
Institut fuer Analysis und Numerik, Otto-von-Guericke-Universitaet Magdeburg, D-39106 Magdeburg, Germany
Email:
eehan@math.uni-bremen.de
Gerald Warnecke
Affiliation:
Institut fuer Analysis und Numerik, Otto-von-Guericke-Universitaet Magdeburg, D-39106 Magdeburg, Germany
MR Author ID:
261694
Email:
warnecke@ovgu.de
Keywords:
Shock waves,
rarefaction waves,
velocity function,
stationary waves,
resonant waves,
Froude number,
nonuniqueness solutions
Received by editor(s):
June 12, 2012
Published electronically:
June 13, 2014
Additional Notes:
The first author is supported by Micro-Macro-Interactions in structured Media and Particle Systems
Article copyright:
© Copyright 2014
Brown University