Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique

Mathematical Models and Methods in Applied Sciences

Volumn 17, Issue 12, 2007, Pages 2055-2113

  Castro, Manuel J ^a   Milanés, Alberto Pardo ^a   Parés, Carlos ^a  

^a UNIVERSITY OF MÁLAGA   (Spain)

Author keywords

Approximate Riemann solvers; Finite volume method; Godunov methods; High order methods; Nonconservative products; Relaxation methods; Roe methods; Well balanced schemes

Indexed keywords

EID: 37249062775     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021820250700256X     Document Type: Article

References (44)

1. R. Abgrall and S. Kami, A relaxation scheme for the two-layer shallow water system, to appear in the Proceedings of the HYP2006 Conference.
2. 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. Sci. Comput. 25 (2004) 2050-2065.
3. E. Audusse and M. O. Bristeau, A well-balanced positivity preserving second order scheme for shallow water flows on unstructured meshes, J. Comput. Phys. 206 (2005) 311-333.
4. E. Audusse, A multilayer Saint-Venant model: Derivation and numerical validation, Discr. Cont. Dyn. Syst. Ser. B 5 (2005) 189-214.
5. A. Bermúdez and M. E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms, Comp. Fluids 23 (1994) 1049-1071.
6. C. Berthon, Schema nonlinéaire pour l'approximation numérique d'un système hyperbolique non conservatif, C. R. Math. Acad. Sci. Paris 335 (2002) 1069-1072.
7. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources (Birkhäuser, 2004).
8. M. J. Castro, J. M. Gallardo and C. Parés, High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow water systems, Math. Comp. 75 (2006) 1103-1134.
9. M. J. Castro, J. A. Garcia, J. M. González, J. Macías, C. Parés and M. E. Vázquez, Numerical simulation of two layer shallow water flows through channels with irregular geometry, J. Comput. Phys. 195 (2004) 202-235.
10. M. J. Castro, J. M. González and C. Parés, Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme, Math. Mod. Meth. Appl. Sci. 16 (2006) 897-931.
11. M. J. Castro, J. Macías and C. Parés, 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, Math. Mod. Numer. Anal. 35 (2001) 107-127.
12. T. Chacón, A. Domínguez and E. D. Fernández, A family of stable numerical solvers for Shallow Water equations with source terms, Comp. Meth. Appl. Mech. Eng. 192 (2003) 203-225.
13. T. Chacón, A. Domínguez and E. D. Fernández, An entropy correction free solver for non-homegeneous shallow water equations, Math. Mod. Numer. Anal. 37 (2003) 755-772.
14. T. Chacón, A. Domínguez and E. D. Fernández, Asymptotically balanced schemes for non-homoegeneous hyperbolic systems -application to the shallow water equations, C. R. Acad. Sci. Paris, Ser. 1338 (2004) 85-90.
15. M. J. Castro, T. Chacón, E. D. Fernández and C. Parés, On well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems, SIAM J. Sci. Comput. 29 (2007) 1093-1126.
16. A. Chinnaya and A. Y. LeRoux, A new general Riemann solver for the shallow-water equations with friction and topography, available at http://www.math.ntnu.no/conservation/1999/021.html.
17. A. Chinnaya, A. Y. LeRoux and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: The resonance phenomenon, Int. J. Finite 1 (2004) 1-33 (electronic).
18. G. Dal Maso, P. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995) 483-548.
19. T. Gallouët, J.-M. Huérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography, Comp. Fluids 32 (2003) 479-513.
20. P. García-Navarro and M. E. Vázquez-Cendón, On numerical treatment of the source terms in the shallow water equations, Comp. Fluids 29 (2000) 17-45.
21. G. Godinaud, A. Y. LeRoux and M. N. LeRoux, Generation of new solvers involving the source term for a class of nonhomogeneous hyperbolic systems, available at http://www.math.ntnu.no/conservation/2000/029.html.
22. 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.
23. L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms, Math. Mod. Meth. Appl. Sci. 11 (2001) 339-365.
24. 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.
25. J. M. Greenberg, A. Y. LeRoux, R. Baraille and A. Noussair, Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal. 34 (1997) 1980-2007.
26. A. Harten, P. D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983) 35-61.
27. P. G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math. and its Appl., Minneapolis, preprint 593, 1989.
28. P. G. LeFloch, Graph solutions of nonlinear hyperbolic systems, J. Hyper. Diff. Eqns. 2 (2004) 643-689.
29. R. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys. 146 (1998) 346-365.
30. A. Marquina, Local piecewise hyperbolic reconstructions for nonlinear scalar conservation laws, SIAM J. Sci. Comp. 15 (1994) 892-915.
31. M. L. Muñoz and C. Parés, Godunov method for nonconservative hyperbolic systems, Math. Mod. Numer. Anal. 41 (2007) 169-185.
32. S. Noelle, N. Pankratz, G. Puppo and J. R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys. 213 (2006) 474-499.
33. S. Noelle, Y. Xing and C.-W. Shu, High order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. Comput. Phys. 226 (2007) 29-58.
34. C. Parés and M. J. Castro, On the well-balanced property of Roe's method for non-conservative hyperbolic systems. Applications to shallow water systems, Math. Mod. Numer. Anal. 38 (2004) 821-852.
35. C. Parés, Numerical methods for nonconservative hyperbolic systems: A theoretical framework, SIAM J. Numer. Anal. 44 (2006) 300-321.
36. B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38 (2001) 201-231.
37. B. Perthame and C. Simeoni, Convergence of the upwind interface source method for hyperbolic conservation laws, in Proc. of Hyp 2002, eds. T. Y. Hou and E. Tadmor (Springer, 2003).
38. P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys. 43 (1981) 357-371.
39. P. L. Roe, Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, eds. Carasso, Raviart and Serre (Springer, 1986), pp. 41-51.
40. G. Russo, Central schemes and Systems of Balance Laws, in Hyperbolic Partial Differential Equations, Theory, Numerics and Applications, eds. A. Meister and J. Struckmeier (Vieweg, Wiesbaden (D), 2002), Chap. 2.
41. C. W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes, J. Comput. Phys. 77 (1998) 439-471.
42. E. F. Toro and M. E. Vázquez-Cendón, Model hyperbolic systems with source terms: exact and numerical solutions, Proc. of Godunov methods: Theory and Applications, 2000.
43. I. Toumi, A weak formulation of Roe approximate Riemann solver, J. Comput. Phys. 102 (1992) 360-373.
44. A. I. Volpert, The space BV and quasilinear equations, Math. USSR. Sbor. 73 (1967) 225-267.