A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers

Journal of Computational Physics

Volumn 230, Issue 3, 2011, Pages 515-550

  Pelanti, Marica ^a   Bouchut, François ^a,c   Mangeney, Anne ^b  

^a ECOLE NORMALE SUPÉRIEURE   (France)

^b INSTITUT DE PHYSIQUE DU GLOBE DE PARIS   (France)

^c Projet Mathrisk INRIA   (France)

Author keywords

Positivity; Relaxation; Riemann solvers; Shallow flow models; Two phase granular flows; Wet dry fronts

Indexed keywords

CONFINED FLOW; DEGREES OF FREEDOM (MECHANICS); NUMERICAL MODELS; TWO PHASE FLOW;

FLOW MODELLING; GRANULAR FLOWS; POSITIVITY; RELAXATION; RIEMANN SOLVER; SHALLOW FLOW; SHALLOW FLOW MODEL; TWO PHASE; TWO-PHASE GRANULAR FLOW; WET/DRY FRONTS;

GRANULAR MATERIALS;

EID: 78649326445     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.10.001     Document Type: Article

References (59)

1. Pitman E.B., Le L. A two-fluid model for avalanche and debris flows. Philos. Trans. Roy. Soc. A 2005, 363:1573-1601.
2. Pirulli M., Mangeney A. Result of back-analysis of the propagation of rock avalanches as a function of the assumed rheology. Rock Mech. Rock Eng. 2008, 74(1):59-84.
3. Kuo C.Y., Tai Y.C., Bouchut F., Mangeney A., Pelanti M., Chen R.F., Chang K.J. Simulation of Tsaoling landslide, Taiwan, based on Saint Venant equations over general topography. Eng. Geol. 2009, 104:181-189.
4. Favreau P., Mangeney A., Lucas A., Crosta G., Bouchut F. Numerical modeling of landquakes. Geophys. Res. Lett. 2010, 37:L15305. 10.1029/2010GL043512.
5. Iverson R.M. The physics of debris flows. Rev. Geophys. 1997, 35:245-296.
6. A. Mangeney, F. Bouchut, N. Thomas, J.-P. Vilotte, M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits, J. Geophys. Res. 112, F02017, doi:10.1029/2006JF000469.
7. N. Mangold, A. Mangeney, V. Migeon, V. Ansan, A. Lucas, D. Baratoux, F. Bouchut, Sinuous gullies on Mars: frequency, distribution, and implications for flow properties, J. Geophys. Res., in press doi:10.1029/2009JE003540.
8. Lucas A., Mangeney A. Mobility and topographic effects for large Valles Marineris landslides on Mars. Geophys. Res. Lett. 2007, 34:l10201.
9. Pelanti M., Bouchut F., Mangeney A., Vilotte J.-P. Numerical modeling of two-phase gravitational granular flows with bottom topography. Hyperbolic Problems: Theory, Numerics and Applications, Proceedings of the Eleventh International Conference on Hyperbolic Problems (Lyon, France, 2006) 2008, 825-832. Springer. S. Benzoni-Gavage, D. Serre (Eds.).
10. Pelanti M., Bouchut F., Mangeney A. A Roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM-Math. Model. Numer. 2008, 42:851-885.
11. Godlewski E., Raviart P.-A. Numerical Approximation of Hyperbolic Systems of Conservation Laws 1996, Springer-Verlag, New York.
12. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics 1997, Springer-Verlag, Berlin, Heidelberg.
13. LeVeque R.J. Finite Volume Methods for Hyperbolic Problems 2002, Cambridge University Press.
14. Toro E.F. Shock-Capturing Methods for Free-Surface Shallow Flows 2001, Wiley.
15. Hervouet J.-M. Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method 2007, Wiley.
16. Godunov S.K. A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 1959, 47:271-306.
17. Harten A., Lax P.D., van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 1983, 25:35-61.
18. Suliciu I. On modelling phase transitions by means of rate-type constitutive equations, shock wave structure. Int. J. Eng. Sci. 1990, 28:829-841.
19. Suliciu I. Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations. Int. J. Eng. Sci. 1992, 30:483-494.
20. Bouchut F. Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources 2004, Birkhäuser-Verlag.
21. Berthon C., Marche F. A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 2008, 30:2587-2612.
22. George D.L. Augmented Riemann solvers for the shallow water equations with steady states and inundation. J. Comput. Phys. 2008, 227(6):3089-3113.
23. LeVeque R.J., Pelanti M. A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 2001, 172:572-591.
24. Castro M.J., González-Vida J.M., Parés C. Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme. Math. Model. Methods Appl. Sci. 2006, 16(6).
25. Gallardo J.M., Parés C., Castro M. On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 2007, 227:574-601.
26. Buffard T., Gallouët T., Hérard J.-M. Un schéma simple pour les équations de Saint-Venant. C.R. Acad. Sci. Paris, Série I 1998, 323:386-390.
27. Gallouët T., Hérard J.-M., Seguin N. On the use of symmetrizing variables for vacuums. Calcolo 2003, 40:163-194.
28. Gallouët T., Hérard J.-M., Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 2003, 32:479-513.
29. Jin S., Xin Z.P. The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 1995, 48:235-276.
30. Liu T.P. Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 1987, 108:153-175.
31. Coquel F., Perthame B. Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 1998, 35:2223-2249.
32. Gallouët T., Masella J. Un schéma de Godunov approché. C.R. Acad. Sci. Paris, Série I 1996, 323:77-84.
33. Masella J.M., Faille I., Gallouët T. On an approximate Godunov scheme. Int. J. Comput. Fluid Dyn. 1999, 12:133-149.
34. Roe P.L. Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 1981, 43:357-372.
35. L.H. Le, New models for geophysical mass flows, Ph.D. Thesis, State University of New York at Buffalo, 2006.
36. LeVeque R.J. Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 1997, 131:327-353.
37. LeVeque R.J. Numerical Methods for Conservation Laws 1990, Birkhäuser-Verlag.
38. Bale D., LeVeque R.J., Mitran S., Rossmanith J.A. A wave-propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 2002, 24:955-978.
39. Toumi I. A weak formulation of Roe's approximate Riemann solver. J. Comput. Phys. 1992, 102:360-373.
40. Parés C., Castro M.J. On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM-Math. Model. Numer. 2004, 38:821-852.
41. Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 2006, 44:300-321.
42. Einfeldt B., Munz C.D., Roe P.L., Sjögreen B. On Godunov-type methods near low densities. J. Comput. Phys. 1991, 92:273-295.
43. Harten A., Hyman J.M. Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 1983, 50:235-269.
44. LeVeque R.J., George D.L. High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. Advances in Coastal and Ocean Engineering 2008, vol. 10:43-73. World Scientific. P.L.-F. Liu, H. Yeh, C. Synolakis (Eds.).
45. Buffard T., Gallouët T., Hérard J.-M. A naïve Riemann solver to compute a non-conservative hyperbolic system. International Series of Numerical Mathematics 1999, vol. 129:129-138. Birkhäuser-Verlag.
46. Declercq E., Forestier A., Hérard J.-M. Comparaison de solveurs numériques pour simuler la turbulence compressible. C.R. Acad. Sci. Paris, Série I 2000, 331:1011-1016.
47. Ovsyannikov L.V. Two-layer shallow water models. Prikl. Mekh. Tekh. Fiz. 1979, 2:3-14.
48. Vreugdenhil C.B. Two-layer shallow-water flow in two dimensions, a numerical study. J. Comput. Phys. 1979, 33:169-184.
49. Castro M.J., Macías J., Parés C. 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-Math. Model. Numer. 2001, 35:107-127.
50. Abgrall R., Karni S. Two-layer shallow water systems: a relaxation approach. SIAM J. Sci. Comput. 2009, 31:1603-1627.
51. Buffard T., Gallouët T., Hérard J.-M. A sequel to a rough Godunov scheme: application to real gases. Comput. Fluids 2000, 29:813-847.
52. Maso G.D., LeFloch P.G., Murat F. Definition and weak stability of nonconservative products. J. Math. Pure Appl. 1995, 74:483-548.
53. Castro M.J., LeFloch P., Muñoz-Ruiz M.L., Parés C. Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys. 2008, 227:8107-8129.
54. Abgrall R., Karni S. A comment on the computation of non-conservative products. J. Comput. Phys. 2010, 229(8):2759-2763.
55. R.J. LeVeque, clawpack, http://www.amath.washington.edu/claw.
56. Pelanti M., Quartapelle L., Vigevano L. Low dissipation entropy fix for positivity preserving Roe's scheme. Godunov Methods: Theory and Applications 2001, 685-690. Kluwer/Plenum Academic Press. E.F. Toro (Ed.).
57. Dumbser M., Castro M., Parés C., Toro E.F. ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows. Comput. Fluids 2009, 38:1731-1748.
58. Castro M.J., Pardo A., Parés C., Toro E.F. On some fast well-balanced first order solvers for nonconservative systems. Math. Comput. 2010, 79:1427-1472.
59. Audusse E., Bouchut F., Bristeau M.-O., Klein R., Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 2004, 25:2050-2065.