A multilayer Saint-venant model: Derivation and numerical validation

Discrete and Continuous Dynamical Systems - Series B

Volumn 5, Issue 2, 2005, Pages 189-214

  Audusse, E ^a  

^a INRIA ROCQUENCOURT   (France)

Author keywords

Free surface Navier Stokes equations; Hydrostatic approximation; Multilayer model; Saint Venant system; Shallow water assumption; Viscosity

Indexed keywords

EID: 18144362100     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2005.5.189     Document Type: Article

References (28)

1. E. Audusse, F. Bouchut, M. O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sc. Comp., 25 (2004), no. 6, 2050-2065.
2. E. Audusse, M. O. Bristeau and B. Perthame, Kinetic schemes for Saint-Venant equations with source terms on unstructured grids, INRIA Report, RR-3989 (2000), http://www.inria.fr/RRRT/RR-3989.html
3. A. Bermudez and E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23 (1994), no. 8, 1049-1071.
4. Besson O. and Laydi M.R., Some estimates for the anisotropic Navier-Stokes equations and for the hydrostatic approximation, RAIRO M2AN, 26 (1992), no. 7, 855-865.
5. 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, C. R. Math. Acad. Sci. Paris, 336 (2003), no. 6. 531-536.
6. M. Castro, J. Macias and C. Pares, 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. M2AN Math. Model. Numer. Anal., 35 (2001), no. 1, 107-127.
7. M. J. Castro , J. A. Garcia-Rodriguez, J. M. Gonzanlez-Vida, J. Macias, C. Pares and E. Vazquez-Cendon, Numerical Simulation of two layer shallow water flows through channels with irregular geometry, JCP, 198 (2004), 202-235.
8. M. J. Castro and C. Pares, On the well balanced property of Roe's method for nonconservative hyperbolic systems. Applications to shallow water systems, M2AN, 38 (2004), no. 5, 821-852.
9. P. Causin, E.Miglio and F. Saleri, Algebraic factorizations for 3D non-hydrostatic free surface flows, Comput. Vis. Sci., 5 (2002), no. 2, 85-94.
10. T. Chacon Rebollo and F. Guillen-Gonzalez, An intrinsic analysis of existence of solutions for the hydrostatic approximation of Navier-Stokes equations, C. R. Acad. Sci. Paris, Serie 1, 330 (2000), 841-846.
11. C. M. Dafermos, "Hyperbolic conservation laws in continuum physics", Springer Verlag, GM 325 (1999).
12. S. Ferrari and F. Saleri, A new two dimensionnal shallow water model including pressure term and slow varying bottom topography, M2AN 38 (2004), no. 2, 211-234.
13. L. Fontana, E. Miglio, A. Quarteroni and F. Saleri, A finite element method for 3D hydrostatic water flows, Comput. Vis. Sci., 2 (1999), no. 2, 85-93.
14. J. F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water ; Numerical validation, Discrete Cont. Dyn. Syst. Ser. B, 1 (2001), no. 1 89-102.
15. A. E. Gill, Atmosphere Ocean Dynamics, International Geophysics Series Vol. 30, Academic Press Inc. (1982).
16. E. Godlewski, P.-A. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, 118, Springer-Verlag, New York, 1996.
17. J. M. Greenberg and A. Y. LeRoux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16.
18. F. Guillen-Gonzalez, N. Masmoudi and R. Bellido, Anisotropic estimates and strong solutions of the primitive equations, Diff. Int. Equ., 14 (2001), no. 11, 1381-1408.
19. J. M. Hervouet, A high resolution SD dam break model using parallelization, Hydrological Processes, 14 (2000), no. 13, 2221-2230.
20. J. M. Hervouet, J. M. Janin, F. Lepeintre and P. Pechon, TELEMAC 3D: a finite element software to solve the 3D free surface flow problems, International Conference on Hydroscience and Engineering, Washington, USA, 1993.
21. E. Lazzaroni, Approssimazione numerica di modelli multistrato per fluidi a superficia libera (in Italian), Universita degli studi di Milano, Tesi di laurea, 2002.
22. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.
23. P. L. Lions, Mathematical Topics in Fluid Mechanics Vol. 1, Oxford University Press, 1996.
24. E. Miglio, A. Quarteroni and F. Saleri, Finite element approximation of quasi-3D shallow water equations, Comput. Methods Appl. Mech. Engrg., 174 (1999), no. 3-4, 355-369.
25. B. Perthame, Kinetic formulations of conservation laws, Oxford University Press, 2002.
26. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), 201-231.
27. A. J. C. de Saint-Venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit (in french), C. R. Acad. Sc., Paris, 73 (1871), 147-154.
28. G. B. Whitham, Linear and non-linear waves, John Wiley and Sons Inc., New York, 1999.