Finite-volume solvers for a multilayer Saint-Venant system
Audusse, Emmanuel ; Bristeau, Marie-Odile
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007), p. 311-320 / Harvested from The Polish Digital Mathematics Library
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We consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to D hydrostatic Navier-Stokes equations.

Publié le : 2007-01-01
EUDML-ID : urn:eudml:doc:207838 
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     title = {Finite-volume solvers for a multilayer Saint-Venant system},
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     year = {2007},
     pages = {311-320},
     zbl = {1152.35305},
     language = {en},
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Audusse, Emmanuel; Bristeau, Marie-Odile. Finite-volume solvers for a multilayer Saint-Venant system. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) pp. 311-320. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv17i3p311bwm/

[000] Audusse E. (2005): A multilayer Saint-Venant model. Discrete and Continuous Dynamical Systems, Series B, Vol.5, No.2, pp.189-214. | Zbl 1075.35030

[001] Audusse E. and Bristeau M.O. (2005): A well-balanced positivy preserving second order scheme for shallow water flows on unstructured meshes. Journal of Computational Physics, Vol.206, pp.311-333. | Zbl 1087.76072

[002] Audusse E., Bristeau M.O. and Decoene A. (2006a): 3D free surface flows simulations using a multilayer Saint-Venant model - Comparisons with Navier-Stokes solutions. Proceedings of the 6-th European Conference Numerical Mathematics and Advanced Applications, ENUMATH 2005, Santiago de Compostella, Spain, pp.181-189. | Zbl 1124.76037

[003] Audusse E., Klein R. and Owinoh A. (2006b): Conservative and well-balanced discretizations for shallow water flows on rotational domains. Proceedings of the 77th Annual Scientific Conference Gesellschaft fur Angewandte Mathematik und Mechanik, GAMM 2006, Berlin, Germany.

[004] Audusse E., Bristeau M.O. and Decoene A. (2007): Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model. (submitted). | Zbl 1139.76036

[005] Bermudez A. and Vazquez M.E. (1994): Up wind methods for hyperbolic conservation laws with source terms. Computers and Fluids, Vol.23, No.8, pp.1049-1071. | Zbl 0816.76052

[006] Benkhaldoun F., Elmahi I. and Monthe L.A. (1999): Positivy preserving finite volume Roe schemes for transport-diffusion equations. Computer Methods in Applied Mechanics and Engineering, Vol.178, pp.215-232. | Zbl 0967.76063

[007] Bouchut F. (2002): An introduction to finite volume methods for hyperbolic systems of conservation laws with source. Ecole CEA - EDF - INRIA Ecoulements peu profonds à surface libre, Octobre 2002, INRIA Rocquencourt http://www.dma.ens.fr/fbouchut/publications/fvcours.ps.gz

[008] Bouchut F., Le Sommer J. and Zelin V. (2004): Frontal geostrophic adjustment and nonlinear wave phenomena in one dimensional rotating shallow water. Part 2: High-resolution numerical simulations. Journal of Fluid Mechanics, Vol.513, pp.35-63. | Zbl 1067.76093

[009] Bristeau M.O. and Coussin B. (2001): Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes. INRIA Report, Vol.4282, http://www.inria.fr/RRRT/RR-4282.html

[010] Castro M., Macias J. and Pares C. (2001): A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: Mathematical Modelling and Numerical Analysis, Vol.35, pp.107-127. | Zbl 1094.76046

[011] Einfeldt B., Munz C.D., Roe P.L. and Slogreen B. (1991): On Godunov type methods for near low densies. Journal of Computational Physics, Vol.92, pp.273-295. | Zbl 0709.76102

[012] Ferrari S. and Saleri F. (2004): A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography. ESAIM: Mathematical Modelling and Numerical Analysis, Vol.38, No.2, pp.211-234. | Zbl 1130.76329

[013] George D. (2004): Numerical Approximation of the Nonlinear Shallow Water Equations with Topography and Dry Beds: A Godunov-Type Scheme. M.Sc. Thesis, University of Washington.

[014] Gerbeau J.-F. and Perthame B. (2001): Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation. Discrete and Continuous Dynamical Systems, Ser. B,Vol.1, No.1, pp.89-102. | Zbl 0997.76023

[015] Godlewski E. and Raviart P.-A. (1996): Numerical Approximation of Hyperbolic Systems of Conservation Laws. New York: Springer-Verlag. | Zbl 0860.65075

[016] Godunov S.K. (1959): A difference method for numerical calculation of discontinuous equations of hydrodynamics. Matematicheski Sbornik, pp.271-300, (in Russian). | Zbl 0171.46204

[017] Harten A., Lax P.D. and Van Leer B. (1983): On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, Vol.25, pp.35-61. | Zbl 0565.65051

[018] Hervouet J.M. (2003): Hydrodynamique des écoulements à surface libre; Modélisation numérique avec la méthode des éléments finis. Paris: Presses des Ponts et Chaussées, (in French).

[019] Khobalatte B. (1993): Resolution numerique des equations de la mécanique des fluides par des methides cinétiques. Ph.D. Thesis, Université P. and M. Curie (Paris 6), (in French).

[020] Le Veque R.J. (1992): Numerical Methods for Conservation Laws. Basel: Birkhauser.

[021] Lions P.L., Perthame B. and Souganidis P.E. (1996): Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Communications on Pure and Applied Mathematics, Vol.49, No.6, pp.599-638. | Zbl 0853.76077

[022] Perthame B. (2002): Kinetic Formulations of Conservation Laws. Oxford: Oxford University Press. | Zbl 1030.35002

[023] Perthame B. and Simeoni C. (2001): A kinetic scheme for the Saint-Venant system with a source term. Calcolo,Vol.38, No.4, pp.201-231. | Zbl 1008.65066

[024] Roe P.L. (1981): Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics, Vol.43, pp.357-372. | Zbl 0474.65066

[025] de Saint-Venant A.J.C. (1971): Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit (in French).Comptes Rendus de l'Académie des Sciences, Paris, Vol.73, pp.147-154 | Zbl 03.0482.04