Numerical methods for nonconservative hyperbolic systems: A theoretical framework

SIAM Journal on Numerical Analysis

Volumn 44, Issue 1, 2006, Pages 300-321

  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

APPROXIMATION ALGORITHMS; FINITE VOLUME METHOD; MATHEMATICAL OPERATORS; NUMERICAL METHODS; PROBLEM SOLVING;

APPROXIMATE RIEMANN SOLVERS; GODUNOV METHODS; HIGH ORDER METHODS; NONCONSERVATIVE PRODUCTS; RELAXATION METHODS; ROE METHODS; WELL BALANCED SCHEMES;

LINEAR SYSTEMS;

EID: 33847668045     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/050628052     Document Type: Article

References (41)

1. F. ALOUGES AND B. MERLET, Approximate shock curves of nonconservative hyperbolic systems in one space dimension, J. Hyperbolic Differ. Equ., 1 (2004), pp. 769-788.
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. Comp., 25 (2004), pp. 2050-2065.
3. A. BERMÚDEZ AND M. E. VAZQUEZ, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049-1071.
4. S. BIANCHINI AND A. BRESSAN, Vanishing viscosity solutions of nonlinear hyperbolic systems, Ann. of Math., 161 (2005), pp. 223-342.
5. F. BOUCHUT, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, Basel, Switzerland, 2004.
6. F. BOUCHUT AND F. JAMES, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), pp. 891-933.
7. 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, M2AN Math. Mod. Numer. Anal., 35 (2001), pp. 107-127.
8. M. J. CASTRO, J. M GALLARDO, AND C. PARÉS, Finite volume schemes based on WENO reconstruction of states for solving nonconservative hyperbolic systems. Applications to shallow water systems, Math. Comp., 2005 to appear.
9. J. J. CAURET, J. F. COLOMBEAU, AND A. Y. LEROUX, Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, J. Math. Anal. Appl., 139 (1989), pp. 552-573.
10. T. CHACÓN, A. DOMÍNGUEZ, AND E. D. FERNÁNDEZ, A family of stable numerical solvers for shallow water equations with source terms, Comput. Methods Appl. Mech. Eng., 192 (2003), pp. 203-225.
11. T. CHACÓN, A. DOMÍNGUEZ, AND E. D. FERNÁNDEZ, Asymptotically balanced schemes for nonhomogeneous hyperbolic systems-application to the shallow water equations, C.R. Math. Acad. Sci. Paris, 338 (2004), pp. 85-90.
12. T. CHACÓN, E. D. FERNÁNDEZ, M. J. CASTRO, AND C. PARÉS, On well-balanced finite volume methods for nonhomogeneous nonconservative hyperbolic systems, preprint, 2005.
13. J. F. COLOMBEAU AND A. HEIBIG, Nonconservative products in bounded variation functions, SIAM J. Math. Anal., 23 (1992), pp. 941-949.
14. G. DAL MASO, P. G. LEFLOCH, AND F. MURAT, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), pp. 483-548.
15. F. DE VUYST, Schémas Nonconservatifs et Schémas Cinétiques Oour la Simulation Numérique D'écoulements Hypersoniques Non Visqueux en Déséquilibre Thermochimique, Thése de Doctorat de l'Université Paris VI, Paris, France, 1994.
16. S. K. GODUNOV, A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), pp. 357-393.
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), pp. 135-159.
18. L. GOSSE, A well-balanced scheme using nonconservative products designed for hyperbolic systems of conservation laws with source terms, Math. Models Methods Appl. Sci., 11 (2001), pp. 339-365.
19. L. GOSSE, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp., 71 (2002), pp. 553-582.
20. G. GUERRA, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differential Equations, 206 (2004), pp. 438-469.
21. A. HARTEN, P. D. LAX, AND B. VAN LEER, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25 (1983), pp. 35-61.
22. T. HOU AND P. G. LEFLOCH, Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp., 62 (1994), pp. 497-530.
23. 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), pp. 1-16.
24. 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), pp. 1980-2007.
25. P. G. LEFLOCH, Propagating phase boundaries. Formulation of the problem and existence via the Glimm method, Arch. Rational Mech. Anal., 123 (1993), pp. 153-197.
26. P. G. LEFLOCH AND A. E. TZAVARAS, Representation of weak limits and definition of nonconseruative products, SIAM J. Math. Anal., 30 (1999), pp. 1309-1342.
27. R. LEVEQUE, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm, J. Comput. Phys., 146 (1998), pp. 346-365.
28. R. LEVEQUE, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, UK, 2002.
29. C. PARÉS AND M. J. CASTRO, On the well-balanced property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 821-852.
30. C. PARÉS, J. M. GALLARDO, M. L. MUÑOZ, AND M. J. CASTRO, Godunov's method for nonconservative hyperbolic systems. Application to linear balance laws, preprint, 2005.
31. B. PERTHAME AND C. SIMEONI, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201-231.
32. B. PERTHAME AND C. SIMEONI, Convergence of the upwind interface source method for hyperbolic conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications, Thou and Tadmor, ed., Springer, Berlin, 2003.
33. P. A. RAVIART AND L. SAINSAULIEU, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem, Math. Models Methods Appl. Sci., 5 (1995), pp. 297-333.
34. J. P. RAYMOND, A new definition of nonconservative products and weak stability results, Boll. Un. Mat. Ital. B, 10 (1996), pp. 681-699.
35. P. L. ROE, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357-372.
36. P. L. ROE, Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proceedings of the Conference on Hyperbolic Problems, Carasso, Raviart, and Serre, eds., Springer, 1986, pp. 41-51.
37. H. TANG, T. TANG, AND K. XU, A gas-kinetic scheme for shallow-water equations with source terms, Z. Angew. Math. Phys., 55 (2004), pp. 365-382.
38. T. TANG AND Z. H. TENG, Error bounds for fractional step methods for conservation laws with source terms, SIAM J. Numer. Anal., 32 (1995), pp. 110-127.
39. E. F. TORO, Shock-Capturing Methods for Free-Surface Shallow Flows, Wiley, Chichester, UK, 2001.
40. I. TOUMI, A weak formulation of Roe's approximate Riemann solver, J. Comput. Phys., 102 (1992), pp. 360-373.
41. A. I. VOLPERT, Spaces BV and quasilinear equations, Math. USSR Sbornik, 73 (1967), pp. 255-302.