On well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems

SIAM Journal on Scientific Computing

Volumn 29, Issue 3, 2007, Pages 1093-1126

  Díaz, M J Castro ^a   Rebollo, T Chacón ^b   Fernandez Nieto E D ^b   Parés, Carlos ^a  

^a UNIVERSITY OF MÁLAGA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Shallow water; Source terms; Two layer flows; Upwinding; Well balanced finite volume method

Indexed keywords

APPROXIMATION THEORY; NUMERICAL METHODS; PROBLEM SOLVING; THEOREM PROVING;

SHALLOW WATER; TWO-LAYER FLOWS; WELL-BALANCED FINITE VOLUME METHOD;

FINITE VOLUME METHOD;

EID: 37249012023     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040607642     Document Type: Article

References (21)

1. A. BERMÚDEZ AND M. E. VÁZQUEZ CENDÓN, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049-1071.
2. 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. Model. Numer. Anal., 35 (2001), pp. 107-127.
3. M. J. CASTRO, J. MACÍAS, C. PARÉ S, J. A. RUBAL, AND M. E. VÁZQUEZ, Two-layer numerical model for solving exchange flows through channels with irregular geometry, in Proceedings of ECCOMAS 2001, Swansea, Wales, UK, 2001, pp. 241-247.
4. M. J. CASTRO, J. A. GARCÍA, J. M. GONZÁLEZ, J. MACÍAS, C. PARÉS, AND M. E. VÁZQUEZ, Numerical simulation of two-layer shallow water flows in channels with irregular geometry, J. Comput. Phys., 195 (2004), pp. 202-235.
5. T. CHACÓN REBOLLO, A. DOMÍNGUEZ DELGADO, AND E. D. FERNÁNDEZ NIETO, A family of stable numerical solvers for shallow water equations with source terms, Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 203-225.
6. T. CHACÓN REBOLLO, A. DOMÍNGUEZ DELGADO, AND E. D. FERNÁNDEZ NIETO, An entropy correction-free solver for non-homogeneous shallow-water equations, M2AN Math. Model. Numer. Anal., 37 (2003), pp. 755-772.
7. T. CHACÓN REBOLLO. A. DOMÍNGUEZ DELGADO, AND E. D. FERNÁNDEZ NIETO, Asymptotically balanced schemes for non-homogeneous hyperbolic systems. Application to shallow water equations, C R. Math. Acad. Sci. Paris, 338 (2004), pp. 85-90.
8. G. DAL MASO, P. G. LEFLOCH, AND F. MURAT, Definition and weak stability of non-conservative products, J. Math. Pures Appl., 74 (1995), pp. 483-548.
9. D. FARMER AND L. ARMI, Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, J. Fluid Mech., 164 (1986), pp. 53-76.
10. E. D. FERNÁNDEZ NIETO, Aproximación numérica de leyes de conservación hiperbólicas no homogéneas. Aplicación a las ecuaciones de Aguas Someras, Ph.D. thesis, Universidad de Sevilla, Spain, 2003.
11. A. C FOWLER, Mathematical Model in the Applied Sciences, Cambridge University Press, Cambridge, UK, 1997.
12. E. GODLEWSKI AND P. A. RAVIART, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer-Verlag, New York, 1996.
13. 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.
14. A. KURGANOV AND D. LEVY, Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397-425.
15. R. J. LEVEQUE, Numerical Methods for Conservation Laws, Birkhäuser-Verlag, Basel, 1990.
16. 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, M2AN Math. Model. Numer. Anal., 38 (2004), pp. 821-852.
17. B. PERTHAME AND C SIMEONI, A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38 (2001), pp. 201-231.
18. B. PERTHAME AND C. SIMEONI, Convergence of the upwind interface source method for hyperbolic conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications. Proceedings of the 9th International Conference on Hyperbolic Problems, T. Thou and E. Tadmor, eds., Springer-Verlag, New York, 2003, pp. 61-78.
19. P. L. ROE, Upwind differencing schemes for hyperbolic conservation laws with source terms, in Nonlinear Hyperbolic Problems, C. Carraso, P.-A. Raviart, and D. Serre, eds., Lecture Notes in Math. 1270, Springer-Verlag, Berlin, 1986, pp. 41-51.
20. J. B. SCHIJF AND J. C. SCHONFBLD, Theoretical considerations on the motion of salt and fresh water, in Proceedings of the Minnesota International Hydraulic Convention, Joint meeting IAHR and Hyd. Div. ASCE., 1953, pp. 321-333.
21. A. I. VOLPERT, The space BV and quasilinear equations, Math. USSR Sb., 73 (1967), pp. 225-267.