A sign matrix based scheme for non-homogeneous PDE's with an analysis of the convergence stagnation phenomenon

Journal of Computational Physics

Volumn 226, Issue 2, 2007, Pages 1753-1783

  Sahmim, Slah ^a   Benkhaldoun, Fayssal ^b   Alcrudo, Francisco ^c  

^a University of Sfax   (Tunisia)

^b University Sorbonne Paris Nord   (France)

^c UNIVERSITY OF ZARAGOZA   (Spain)

Author keywords

Euler equations; Finite volumes; Riemann problems; Shallow water equations; Source terms; SRNH scheme; Two phase flow

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; EULER EQUATIONS; FINITE VOLUME METHOD; JACOBIAN MATRICES;

EULER'S EQUATION; FINITE-VOLUME; NON-HOMOGENEOUS; RIEMANN PROBLEM; SHALLOW WATER EQUATIONS; SIGN MATRIX; SOURCE TERMS; SRNH SCHEME; STAGNATION PHENOMENON; TWO PHASES FLOW;

TWO PHASE FLOW;

EID: 34548774022     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2007.06.017     Document Type: Article

References (63)

1. Abgrall R., and Saurel R. Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186 (2003) 361-396
2. Alcrudo F., and Garcia-Navarro P. A high resolution Godunov-type scheme in finite volumes for the 2d shallow water equation. Int. J. Numer. Methods Fluids 16 (1993) 489-505
3. Alcrudo F., and Benkhaldoun F. Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput Fluids 30 6 (2001) 643-671
4. F. Alcrudo and F. Benkhaldoun, On the convergence of some numerical schemes for the solution of shallow water equations with singular source terms, in: R. Herbin, D. Kroner (Eds.), FVCA3, HPS, 2002, pp. 269-276.
5. Andrianov N., and Warnecke G. On the solution to the Riemann problem for compressible duct flow. SIAM J. Appl. Math. 64 3 (2004) 878-901
6. F. Alouges, Matrice signe et systes hyperboliques, rapport CMLA. http://mahery.math.u-psud.fr/{reversed tilde}alouges/.
7. Audusse E., Bouchut F., Bristeau M.O., Klein R., and Perthame B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 6 (2004) 2050-2065
8. F. Benkhaldoun, Analysis and validation of a new finite volume scheme for nonhomogeneous systems, in: R. Herbin, D. KrOner (Eds.), FVCA3, HPS, 2002, pp. 269-276.
9. Benkhaldoun F., and Quivy L. A non homogeous Riemann solver for shallow water and two-phase flows. Flow Turbulence Combust. 76 (2006) 391-402
10. F. Benkhaldoun, I. Elmahi, M. Seaid, Well-balanced finite volume schemes for pollutant transport on unstructured meshes, J. Comput. Phys, in press, doi:10.1016/j.jcp.2007.04.005.
11. Bermúdez A., and Vázquez M.E. Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 8 (1994) 1049
12. Bermúdez A., Dervieux A., Desideri J.A., and Vázquez M.E. Upwind schemes for the two dimensional shallow water equations with variable depth using unstructured meshes. Comput. Methods Appl. Mech. Eng. 155 (1994) 49
13. C. Bon, Modélisation et simulations numeriques d'écoulements hydrauliques et de ruissellement en topographic quelconque, PhD thesis, Université Bordeaux I, France, 1997.
14. M. Bouker, Modélisation numérique multidimentionnelle d'écoulements diphasiques liquide-gaz en régimes transitoire et permanant: méthodes et applications, PhD Thesis, Electricité de France, 1998.
15. F. Chatelen, "Valeurs propres des matrices, M.A.M., MASSON, 1988.
16. A. Chinnayya, A.Y. LeRoux, A new general Riemann solver for the shallow water equations with friction and topography, Available from the conservation law preprint server: http://www.math.ntnu.no/conservation/, 1999.
17. A. Chinnayya, A.Y. LeRoux and N. Seguin, (2004), "A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomena, Int. J. Finite Volumes, http://averoes.math.univ-paris13.fr, vol.1, No. 1.
18. Coquel F., El Amine K., Godlewski E., Perthame B., and Rascle P. A numerical method using upwind schemes for the resolution of two-phase flows. J. Comput. Phys. 136 (1997) 272-288
19. Cortes J., Debussche A., and Toumi I. A density perturbation method to study the eigenstructure of two-phase flow equation systems. J. Comput. Phys. (1998) 463-484
20. Dal Maso G., Lefloch P., and Murat P. Definition and weak stability of a non conservative product. J. Math Pure Appl. 74 (1995) 483-548
21. Depres B. Lax theorem and finite volume schemes. Math. Comput. 73 247 (2003) 1203-1234
22. Gallouët T., and Masella J.M. Un schéma de Godunov approché. C.R. Acad. Sci. Paris, t.323, Série I (1996) 77-84
23. Gallouet T., Hérard J.-M., and Seguin N. Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513
24. Gallouët T., Hérard J.M., and Seguin N. Numerical modeling of two-phase flows using the two-fluid two-pressure approch. Math. Models Methods Appl. Sci. (M3AS) 14 5 (2004) 663-700
25. Ghidaglia J.M., Kumbaro A., and Le Coq G. Une méthode "volumes finis" à flux caractéristiques pour la résolution numérique des systèmes hyperboliques de lois de conservation. C.R. Acad. Sci. Paris Sér. I Math. 322 10 (1996) 981-988
26. Ghidaglia J.M., Kumbaro A., and Le Coq G. On the numerical solution to two fluid models via a cell centered finite volume method. Eur. J. Mech. B. Fluids (2001) 841-867
27. Glaister P. Prediction of supercritical flow in open channels. Comput. Math. Appl. 24 7 (1992) 69-75 October
28. E. Godlewski, P.-A. Raviart, Hyperbolic systems of conservation laws, Mathématiques Appl. (Paris), 3/4, Ellipses, 1991, p. 252.
29. Gosse L., and Leroux A.Y. Un schéma-équilibré adapté aux lois de conservation scalaires non-homogènes. C.R. Acad. Sci. Paris 323 (1996) 543-546
30. Grandotto M., and Obry P. Calculs des écoulements diphasiques dans les échangeurs par une méthode aux éléments finis. Revue Europénne des Eléments Finis 5 1 (1996)
31. M. Grandotto, P. Goldbronn, I. Toumi, P. Fillion, Comparison of finite element and finite volume algorithms for themohydraulic studies of steam generators and nuclear reactors cores, 10th International Conference on Finite Elements in Fluids, Tucson, AZ, USA, 1998.
32. Greenberg J.M., and Le Roux A.Y. A well balanced scheme for the numerical processing of source terms in hyperbolic equation. SIAM. J. Numer. Anal. 33 1 (1996) 1-16
33. Greenberg J.M., LeRoux A.Y., Baraille R., and Noussa ir A. Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980-2007
34. Hinch E.J. Perturbation Methods. Cambridge Texts in Applied Mathematics (1991), Cambridge University Press
35. Hong J.M. An extension of Glimms method to inhomogeneous strictly hyperbolic systems of conservation laws by weaker than weak solutions of the Riemann problem. J. Differ. Eq. 222 (2006) 515-549
36. Ishii M. Thermo-fluid dynamic theory of two-phase flows. Collection de la direction des études et recherches d'Electricité de France, Eyrolle (1975)
37. T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, 1995.
38. Kröner D., and Thanh M.D. Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Num. Anal. 43 2 (2005) 796-824
39. Kurganov A., and Levy D. Central-upwind schemes for the Saint-Venant system. Math. Model. Numer. Anal. (M2AN) 36 3 (2002) 397-425
40. Kumbaro A., Toumi I., and Seignole V. Flux schemes for the numerical simulation of two phase flows. Comput. Fluid Dynam. (2002), Springer pp. 621-626
41. Lahey Jr. R.T. The prediction phase of phase distribution and separation phenomena using two-fluid models. Boiling Heat Transfer (1992), Elsevier Science pp. 85-121
42. Lefloch P.G., and Thanh M.D. The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1 (2003) 763-797
43. A.Y. LeRoux (1998), "Discrétisation des termes sources raides dans les problèmes hyperboliques. In: Systèmes hyperboliques: Nouveaux schémas et nouvelles applications". Ecoles CEA-EDF-INRIA problèmes non linéaires appliqués, INRIA Rocquencourt (France), March 1998 [in French]. Available from: http://www-gm3.univ-mrs.fr/leroue/bublications/ay.leroux.html.
44. LeVeque R.J., and Yee H.C. A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86 (1990) 187
45. LeVeque R.J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365
46. Masella J.-M., Faille I., and Gallouët T. On an approximate Godunov scheme. Int. J. Comput. Fluid Dynam. (1999) 133-149
47. Tseng M.-H. The improved surface gradient method for flows simulation in variable bed topography channel using TVD-MacCormack scheme. Int. J. Numer. Methods Fluids 43 1 (2003) 71-91
48. Perthame B., and Simeoni C. A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201-231
49. V.H. Ransom, Numerical Benchmark Tests, in: G.F. Hewitt, J.M. Delhaye, N. Zuber, (Eds.), Multiphase Science and Technology, vol. 3, Hemisphere Publishing Corporation.
50. Raviart P.A., and Sainsaulieu L. Mathematical and numerical modelling of two-phase flows. In: Glowinski (Ed). Comput Methods Appl. Sci. Eng. (1991), Nova Science Publisher
51. Rebollo T.C., Delgado A.D., and Fernandez Nieto E.D. A family of stable numerical solvers for the shallow water equations with source terms. Comput. Methods Appl. Mech. Eng. 192 (2003) 203-225
52. Roe P.L. Approximate Riemann solvers, parameter vectors and difference dchemes. J. Comput. Phys. 43 (1981) 357-372
53. Roe P.L. Upwind differenced schemes for hyperbolic conservation laws with source terms. Proc. Conf. Hyperbolic Problems (1986) 41
54. Rusanov V.V. The calculation of the interaction of non-stationary shock waves with barriers. JŽ. Vyčisl. Mat. i Mat. Fiz. 1 (1961) 267-279
55. S. Sahmim, Un schéma aux volumes finis avec matrice signe pour les systèmes non homogènes, Ph.D. Thesis, Université Paris 13, 2005.
56. Sainsaulieu L. Finite Volume approximation of two-phase fluid flows based on an approximate Roe-Type Riemann solver. J. Comput. Phys. 121 (1995) 1-28
57. Shi J., Hu C., and Shu C.-W. A technique of treating negative weights in WENO schemes. J. Comput. Phys. 175 (2002) 108-127
58. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. second ed. (1999), Springer-Verlag 624pp
59. Toumi I., and Kumbaro A. An approximate linearized Riemann solver for a two-fluid model. J. Comput. Phys. 124 2 (1996) 286-300
60. Vazquez M.E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148 (1999) 497-526
61. Xing Y., and Shu C.W. High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208 (2005) 206-227
62. Xing Y., and Shu C.W. High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with soure terms. J. Comput. Phys. 214 (2006) 567-598
63. Zhou J.G., Causon D.M., Mingham C.G., and Ingram D.M. The surface gradient method for the treatment of source terms in the shallow-water equations. J. Comput. Phys. 168 1 (2001) 1-25