A well-balanced positivity preserving "second-order" scheme for shallow water flows on unstructured meshes

Journal of Computational Physics

Volumn 206, Issue 1, 2005, Pages 311-333

  Audusse, Emmanuel ^a   Bristeau, Marie Odile ^a  

^a INRIA   (France)

Author keywords

Finite volumes; Hydrostatic reconstruction; Kinetic solver; Positivity preserving scheme; Saint Venant system; Second order extension; Shallow water flow; Well balanced scheme

Indexed keywords

FINITE VOLUME METHOD; FLOW OF WATER; MESH GENERATION;

FINITE-VOLUME; HYDROSTATIC RECONSTRUCTION; KINETIC SOLVERS; POSITIVITY PRESERVING; POSITIVITY PRESERVING SCHEME; PROPERTY; SAINT-VENANT SYSTEM; SECOND-ORDER EXTENSION; SHALLOW-WATER FLOW; WELL- BALANCED SCHEMES;

HYDRAULICS;

EID: 33749016293     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2004.12.016     Document Type: Article

References (42)

1. R. Abgrall, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys. 114 (1) (1994) 45-58.
2. E. Audusse, A multilayer Saint-Venant model, Discrete Cont. Dyn. Syst. Ser. B 5 (2) (2005) 189-214.
3. E. Audusse, F. Bouchut, M.O. Bristeau, R. Klein, B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comp. 25 (6) (2004) 2050-2065.
4. E. Audusse, M.O. Bristeau, Transport of pollutant in shallow water, a two time steps kinetic method, M2AN 37 (2) (2003) 389-416.
5. A. Bermudez, A. Dervieux, J.A. Desideri, M.E. Vazquez, Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. Meth. Appl. Mech. Eng. 155 (1-2) (1998) 49-72.
6. A. Bermudez, M.E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids 23 (8) (1994) 1049-1071.
7. F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources, Frontiers in Mathematics, Birkhäuser, 2004.
8. F. Bouchut, A. Mangeney-Castelnau, B. Perthame, 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 (6) (2003) 531-553.
9. M.O. Bristeau, B. Coussin, Boundary conditions for the shallow water equations solved by kinetic schemes, INRIA Report, 4282, 2001. Available from:.
10. M.O. Bristeau, B. Perthame, Transport of pollutant in shallow water using kinetic schemes, in: ESAIM Proceedings, vol. 10, CEMRACS, 1999, pp. 9-21. Available from: .
11. R. Botchorishvili, B. Perthame, A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, INRIA Report, 3891, 2000. Available from: .
12. T. Chacon Rebollo, A.D. Delgado, E.D.F. Nieto, An entropy-correction free solver for non homogeneous shallow water equations, M2AN 37 (2003) 363-390.
13. T. Chacon Rebollo, A.D. Delgado, E.D.F. Nieto, A family of stable numerical solvers for the shallow water equations with source terms, Comput. Meth. Appl. Math. Eng. 192 (2003) 203-225.
14. A. Chertock, A. Kurganov, On a hybrid final-volume-particle method, M2AN 38 (6) (2004) 1071-1091.
15. S. Ferrari, F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography, M2AN 38 (2) (2004) 211-234.
16. T. Gallouët, J.M. Hérard, N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comput. Fluids 32 (2003) 479-513.
17. L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput. Math. Appl. 39 (2000) 135-159.
18. L. Gosse, A well-balanced scheme using nonconservative products designed for hyperbolic systems of conservation laws with source terms, Math. Mod. Meth. Appl. Sci. 11 (2) (2001) 339-365.
19. J.M. Greenberg, 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.
20. J.-F. Gerbeau, B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation, Discrete Cont. Dyn. Syst. Ser. B 1 (1) (2001) 89-102.
21. E. Godlewski, P.-A. Raviart, Numerical approximations of hyperbolic systems of conservation lawsApplied Mathematical Sciences, vol. 118, Springer, New York, 1996.
22. N. Goutal, F. Maurel, in: Proceedings of the 2nd Workshop on Dam-break Simulation, Note Technique EDF, HE-43/97/016/B, 1997.
23. J.M.N.T. Gray, S. Noelle, Y.C. Tai, Shock waves, dead zones and particle-free regions in rapid granular free-surface flows, J. Fluid Mech. 491 (2003) 161-181.
24. H. Guillard, R. Abgrall, Modélisation Numérique Des Fluides Compressibles, Series in Applied Mathematics, Gauthier-Villars, Paris, 2001 (in French).
25. J.M. Hervouet, Hydrodynamique Des écoulements à Surface Libre; Modélisation Numérique Avec La Méthode Des éléments Finis, Presses des Ponts et Chaussées, 2003 (in French).
26. J.M. Hervouet, A high resolution 2-D dam-break model using parallelization, Hydrol. Process. 14 (2000) 2211-2230.
27. C. Hu, C.W. Shu, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys. 150 (1999) 97-127.
28. S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms, M2AN 35 (2001) 631-645.
29. B. Khobalatte, B. Perthame, Maximum principle of the entropy and second-order kinetic schemes, J. Math. Comp 205 (1994) 119-131.
30. A. Kurganov, D. Levy, Central-upwind schemes for the Saint-Venant system, M2AN 36 (2002) 397-425.
31. R.J. LeVeque, Numerical Methods for Conservation LawsLectures in Mathematics, Birkhäuser, ETH Zurich, 1992.
32. R.J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wavepropagation algorithm, J. Comput. Phys. 146 (1) (1998) 346-365.
33. P.L. Lions, B. Perthame, P.E. Souganidis, Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Commun. Pure Appl. Math. 49 (6) (1996) 599-638.
34. A. Mangeney-Castelnau, J.P. Vilotte, M.O. Bristeau, F. Bouchut, B. Perthame, C. Simeoni, S. Yernini, A new kinetic scheme for Saint-Venant equations applied to debris avalanches, INRIA Report 4646, 2002. Available from: .
35. B. Perthame, Second-order Boltzmann schemes for compressible Euler equations in one or two space dimensions, SIAM J. Numer. Anal. 29 (1) (1992) 1-19.
36. B. Perthame, Kinetic Formulations of Conservation Laws, Oxford University Press, Oxford, 2002.
37. B. Perthame, Y. Qiu, A variant of Van Leer's method for multidimensional systems of conservation laws, J. Comput. Phys. 112 (2) (1994) 370-381.
38. B. Perthame, C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38 (4) (2001) 201-231.
39. P.L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys. 43 (1981) 357-372.
40. G. Russo, Central schemes for balance laws, Hyperbolic Problems: Theory, Numerics, Applications Internat. Ser. Numer. Math., vols. I and II (Magdebourg, 2000), Birkhäuser, Basel, 2001, p. 140, 141, 821-829.
41. 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, C.R. Acad. Sci. Paris 73 (1871) 147-154 (in French).
42. B. Van Leer, Towards the ultimate conservative difference schemes. V. A second-order sequel to the Godunov's method, J. Comput. Phys. 32 (1979) 101-136.