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Godunov method for nonconservative hyperbolic systems

Published online by Cambridge University Press:  26 April 2007

María Luz Muñoz-Ruiz  and
Carlos Parés
María Luz Muñoz-Ruiz
Affiliation:
Dept. Matemática Aplicada, Universidad de Málaga, 29071-Málaga, Spain.
Carlos Parés
Affiliation:
Dept. Análisis Matemático, Universidad de Málaga, 29071-Málaga, Spain.
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Abstract

This paper is concerned with the numerical approximation of Cauchy problems for one-dimensional nonconservative hyperbolic systems. The theory developed by Dal Maso et al. [J. Math. Pures Appl.74 (1995) 483–548] is used in order to define the weak solutions of the system: an interpretation of the nonconservative products as Borel measures is given, based on the choice of a family of paths drawn in the phase space. Even if the family of paths can be chosen arbitrarily, it is natural to require this family to satisfy some hypotheses concerning the relation of the paths with the integral curves of the characteristic fields. The first goal of this paper is to investigate the implications of three basic hypotheses of this nature. Next, we show that, when the family of paths satisfies these hypotheses, Godunov methods can be written in a natural form that generalizes their classical expression for systems of conservation laws. We also study the well-balance properties of these methods. Finally, we prove the consistency of the numerical scheme with the definition of weak solutions: we prove that, under hypothesis of bounded total variation, if the approximations provided by a Godunov method based on a family of paths converge uniformly to some function as the mesh is refined, then this function is a weak solution (related to that family of paths) of the nonconservative system. We extend this result to a family of numerical schemes based on approximate Riemann solvers.

Keywords

Nonconservative hyperbolic systemsGodunov methodwell-balancingapproximate Riemann solvers.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 1 , January 2007 , pp. 169 - 185
Copyright
© EDP Sciences, SMAI, 2007

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References

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser (2004).
A. Bressan, H.K. Jenssen and P. Baiti, An instability of the Godunov Scheme. arXiv:math.AP/0502125 v2 (2005).
Castro, M.J., Macías, J. and Parés, C., Q-Scheme, A for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. Math. Mod. Num. Anal. 35 (2001) 107–127. CrossRef
F. Coquel, D. Diehl, C. Merkle and C. Rohde, Sharp and diffuse interface methods for phase transition problems in liquid-vapour flows, in Numerical Methods for Hyperbolic and Kinetic Problems, IRMA Lectures in Mathematics and Theoretical Physics, Proceedings of CEMRACS 2003.
Dal Maso, G., LeFloch, P.G. and Murat, F., Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74 (1995) 483–548.
F. De Vuyst, Schémas non-conservatifs et schémas cinétiques pour la simulation numérique d'écoulements hypersoniques non visqueux en déséquilibre thermochimique. Ph.D. thesis, University of Paris VI, France (1994).
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996).
Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39 (2000) 135–159. CrossRef
Gosse, L., A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11 (2001) 339–365. CrossRef
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., Lax, P.D. and Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25 (1983) 35–61. CrossRef
Hou, T. and LeFloch, P.G., Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp. 62 (1994) 497–530. CrossRef
Isaacson, E. and Temple, B., Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55 (1995) 625–640. CrossRef
Lax, P.D. and Wendroff, B., Systems of conservation laws. Comm. Pure Appl. Math. 13 (1960) 217–237. CrossRef
P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute Math. Appl., Minneapolis, Preprint 593 (1989).
LeFloch, P.G., Graph solutions of nonlinear hyperbolic systems. J. Hyper. Differ. Equa. 2 (2004) 643–689.
LeFloch, P.G. and Tzavaras, A.E., Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30 (1999) 1309–1342. CrossRef
LeVeque, R.J., 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
Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44 (2006) 300–321. CrossRef
Parés, C. and Castro, M., On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow water systems. Math. Mod. Num. Anal. 38 (2004) 821–852. CrossRef
Volpert, A.I., The space BV and quasilinear equations. Math. USSR Sbornik 73 (1967) 225–267. CrossRef