Finite-volume solvers for a multilayer Saint-Venant system

Emmanuel Audusse; Marie-Odile Bristeau

International Journal of Applied Mathematics and Computer Science (2007)

  • Volume: 17, Issue: 3, page 311-320
  • ISSN: 1641-876X

Abstract

top
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.

How to cite

top

Audusse, Emmanuel, and Bristeau, Marie-Odile. "Finite-volume solvers for a multilayer Saint-Venant system." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 311-320. <http://eudml.org/doc/207838>.

@article{Audusse2007,
abstract = {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.},
author = {Audusse, Emmanuel, Bristeau, Marie-Odile},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {shallow water equations; Saint-Venant system; approximate Riemann solvers; kinetic solver; finite volumes; multilayer model; hyperbolic shallow water models},
language = {eng},
number = {3},
pages = {311-320},
title = {Finite-volume solvers for a multilayer Saint-Venant system},
url = {http://eudml.org/doc/207838},
volume = {17},
year = {2007},
}

TY - JOUR
AU - Audusse, Emmanuel
AU - Bristeau, Marie-Odile
TI - Finite-volume solvers for a multilayer Saint-Venant system
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 3
SP - 311
EP - 320
AB - 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.
LA - eng
KW - shallow water equations; Saint-Venant system; approximate Riemann solvers; kinetic solver; finite volumes; multilayer model; hyperbolic shallow water models
UR - http://eudml.org/doc/207838
ER -

References

top
  1. Audusse E. (2005): A multilayer Saint-Venant model. Discrete and Continuous Dynamical Systems, Series B, Vol.5, No.2, pp.189-214. Zbl1075.35030
  2. 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. Zbl1087.76072
  3. 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. Zbl1124.76037
  4. 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. 
  5. Audusse E., Bristeau M.O. and Decoene A. (2007): Numerical simulations of 3D free surface flows by a multilayer Saint-Venant model. (submitted). Zbl1139.76036
  6. 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. Zbl0816.76052
  7. 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. Zbl0967.76063
  8. 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 
  9. 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. Zbl1067.76093
  10. 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 
  11. 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. Zbl1094.76046
  12. 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. Zbl0709.76102
  13. 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. Zbl1130.76329
  14. 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. 
  15. 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. Zbl0997.76023
  16. Godlewski E. and Raviart P.-A. (1996): Numerical Approximation of Hyperbolic Systems of Conservation Laws. New York: Springer-Verlag. Zbl0860.65075
  17. Godunov S.K. (1959): A difference method for numerical calculation of discontinuous equations of hydrodynamics. Matematicheski Sbornik, pp.271-300, (in Russian). Zbl0171.46204
  18. 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. Zbl0565.65051
  19. 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). 
  20. 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). 
  21. Le Veque R.J. (1992): Numerical Methods for Conservation Laws. Basel: Birkhauser. 
  22. 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. Zbl0853.76077
  23. Perthame B. (2002): Kinetic Formulations of Conservation Laws. Oxford: Oxford University Press. Zbl1030.35002
  24. 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. Zbl1008.65066
  25. Roe P.L. (1981): Approximate Riemann solvers, parameter vectors and difference schemes. Journal of Computational Physics, Vol.43, pp.357-372. Zbl0474.65066
  26. 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 Zbl03.0482.04

Citations in EuDML Documents

top
  1. Ricardo Costa, Gaspar J. Machado, Stéphane Clain, A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation
  2. Ulrik Skre Fjordholm, Siddhartha Mishra, Accurate numerical discretizations of non-conservative hyperbolic systems
  3. Ulrik Skre Fjordholm, Siddhartha Mishra, Accurate numerical discretizations of non-conservative hyperbolic systems
  4. Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, Jacques Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation
  5. Emmanuel Audusse, Marie-Odile Bristeau, Benoît Perthame, Jacques Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.