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On the well-balance property of Roe's method for nonconservative hyperbolic systems. applications to shallow-water systems

Published online by Cambridge University Press:  15 October 2004

Carlos Parés  and
Manuel Castro
Carlos Parés
Affiliation:
Dpto. Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, 29080-Málaga, Spain. grupo@anamat.cie.uma.es.
Manuel Castro
Affiliation:
Dpto. Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, 29080-Málaga, Spain. grupo@anamat.cie.uma.es.
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Abstract

This paper is concerned with the numerical approximations of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The first goal is to introduce a general concept of well-balancing for numerical schemes solving this kind of systems. Once this concept stated, we investigate the well-balance properties of numerical schemes based on the generalized Roe linearizations introduced by [Toumi, J. Comp. Phys.102 (1992) 360–373]. Next, this general theory is applied to obtain well-balanced schemes for solving coupled systems of conservation laws with source terms. Finally, we focus on applications to shallow water systems: the numerical schemes obtained and their properties are compared, in the case of one layer flows, with those introduced by [Bermúdez and Vázquez-Cendón, Comput. Fluids23 (1994) 1049–1071]; in the case of two layer flows, they are compared with the numerical scheme presented by [Castro, Macías and Parés, ESAIM: M2AN35 (2001) 107–127].

Keywords

Nonconservative hyperbolic systemswell-balanced schemesRoe methodsource termsshallow-water systems.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 5 , September 2004 , pp. 821 - 852
Copyright
© EDP Sciences, SMAI, 2004

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References

Andronov, N. and Warnecke, G., On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64 (2004) 878–901.
F. Bouchut, An introduction to finite volume methods for hyperbolic systems of conservation laws with source, in Free surface geophysical flows. Tutorial Notes. INRIA, Rocquencourt (2002).
Bermúdez, A. and Vázquez, M.E., Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23 (1994) 1049–1071. CrossRef
Castro, M.J., Macías, J. and 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: M2AN 35 (2001) 107–127. CrossRef
Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., Parés, C. and Vázquez-Cendón, M.E., Numerical simulation of two-layer Shallow Water flows through channels with irregular geometry. J. Comp. Phys. 195 (2004) 202–235. CrossRef
Chacón, T., Domínguez, A. and Fernández, E.D., A family of stable numerical solvers for Shallow Water equations with source terms. Comp. Meth. Appl. Mech. Eng. 192 (2003) 203–225.
Chacón, T., Domínguez, A. and Fernández, E.D., An entropy-correction free solver for non-homogeneous shallow water equations. ESAIM: M2AN 37 (2003) 755–772.
Chacón, T., Fernández, E.D. and Gómez Mármol, M., A flux-splitting solver for shallow water equations with source terms. Int. Jour. Num. Meth. Fluids 42 (2003) 23–55.
T. Chacón, A. Domínguez and E.D. Fernández, Asymptotically balanced schemes for non-homogeneous hyperbolic systems – application to the Shallow Water equations. C.R. Acad. Sci. Paris, Ser. I 338 (2004) 85–90.
Colombeau, J.F., Le Roux, A.Y., Noussair, A. and Perrot, B., Microscopic profiles of shock waves and ambiguities in multiplications of distributions. SIAM J. Num. Anal. 26 (1989) 871–883. CrossRef
Dal Masso, G., LeFloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548.
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 (2003).
A.C. Fowler, Mathematical Model in the Applied Sciences. Cambridge (1997).
García-Navarro, P. and Vázquez-Cendón, M.E., On numerical treatment of the source terms in the shallow water equations. Comput. Fluids 29 (2000) 17–45. CrossRef
P. Goatin and P.G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, preprint (2003).
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comp. Math. Appl. 39 (2000) 135–159. CrossRef
Gosse, L., A well-balanced scheme using non-conservative products designed for hyperbolic system of conservation laws with source terms. Mat. Mod. Meth. Appl. Sc. 11 (2001) 339–365.
Greenberg, J.M. and LeRoux, A.Y., A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1–16. CrossRef
Greenberg, J.M., LeRoux, A.Y., Baraille, R. and Noussair, A., Analysis and approximation of conservation laws with source terms. SIAM J. Numer. Anal. 34 (1997) 1980–2007. CrossRef
Harten, A. and Hyman, J.M., Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comp. Phys. 50 (1983) 235–269. CrossRef
LeFloch, P.G., Propagating phase boundaries; formulation of the problem and existence via Glimm scheme. Arch. Rat. Mech. Anal. 123 (1993) 153–197.
R. LeVeque, Numerical Methods for Conservation Laws. Birkhäuser (1990).
LeVeque, R., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys. 146 (1998) 346–365. CrossRef
R. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002).
Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38 (2001) 201–231. CrossRef
B. Perthame and C. Simeoni, Convergence of the upwind interface source method for hyperbolic conservation laws, in Proc. of Hyp 2002, Thou and Tadmor Eds., Springer (2003).
Raviart, P.A. and Sainsaulieu, L., A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Mod. Meth. Appl. Sci. 5 (1995) 297–333.
Roe, P.L., Approximate Riemann solvers, parameter vectors and difference schemes. J. Comp. Phys. 43 (1981) 357–371. CrossRef
P.L. Roe, Upwinding difference schemes for hyperbolic conservation laws with source terms, in Proc. of the Conference on Hyperbolic Problems, Carasso, Raviart and Serre Eds., Springer (1986) 41–51.
J.J. Stoker, Water Waves. Interscience, New York (1957).
E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction. Springer-Verlag (1997).
E.F. Toro, Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley (2001).
E.F. Toro and M.E. Vázquez-Cendón, Model hyperbolic systems with source terms: exact and numerical solutions, in Proc. of Godunov methods: Theory and Applications (2000).
Toumi, I., A weak formulation of Roe's approximate Riemann Solver. J. Comp. Phys. 102 (1992) 360–373. CrossRef
M.E. Vázquez-Cendón, Estudio de Esquemas Descentrados para su Aplicación a las Leyes de Conservación Hiperbólicas con Términos Fuente. Ph.D. Thesis, Universidad de Santiago de Compostela (1994).
Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comp. Phys. 148 (1999) 497–526. CrossRef
Volpert, A.I., The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225–267. CrossRef