Godunov method for nonconservative hyperbolic systems

María Luz Muñoz-Ruiz; Carlos Parés

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 1, page 169-185
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Muñoz-Ruiz, María Luz, and Parés, Carlos. "Godunov method for nonconservative hyperbolic systems." ESAIM: Mathematical Modelling and Numerical Analysis 41.1 (2007): 169-185. <http://eudml.org/doc/250025>.

@article{Muñoz2007,
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. },
author = {Muñoz-Ruiz, María Luz, Parés, Carlos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonconservative hyperbolic systems; Godunov method; well-balancing; approximate Riemann solvers.; nonconservative hyperbolic systems; Cauchy problems; weak solutions; systems of conservation laws; consistency},
language = {eng},
month = {4},
number = {1},
pages = {169-185},
publisher = {EDP Sciences},
title = {Godunov method for nonconservative hyperbolic systems},
url = {http://eudml.org/doc/250025},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Muñoz-Ruiz, María Luz
AU - Parés, Carlos
TI - Godunov method for nonconservative hyperbolic systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/4//
PB - EDP Sciences
VL - 41
IS - 1
SP - 169
EP - 185
AB - 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.
LA - eng
KW - Nonconservative hyperbolic systems; Godunov method; well-balancing; approximate Riemann solvers.; nonconservative hyperbolic systems; Cauchy problems; weak solutions; systems of conservation laws; consistency
UR - http://eudml.org/doc/250025
ER -

References

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  1. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser (2004).  
  2. A. Bressan, H.K. Jenssen and P. Baiti, An instability of the Godunov Scheme. arXiv:math.AP/0502125 v2 (2005).  
  3. 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. Math. Mod. Num. Anal.35 (2001) 107–127.  
  4. 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.  
  5. G. Dal Maso, P.G. LeFloch and F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl.74 (1995) 483–548.  
  6. 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).  
  7. E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996).  
  8. L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl.39 (2000) 135–159.  
  9. L. Gosse, 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.  
  10. 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) 1–16.  
  11. 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) 1980–2007.  
  12. A. Harten, P.D. Lax and B. Van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev.25 (1983) 35–61.  
  13. T. Hou and P.G. LeFloch, Why nonconservative schemes converge to wrong solutions: error analysis. Math. Comp.62 (1994) 497–530.  
  14. E. Isaacson and B. Temple, Convergence of the 2 x 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math.55 (1995) 625–640.  
  15. P.D. Lax and B. Wendroff, Systems of conservation laws. Comm. Pure Appl. Math.13 (1960) 217–237.  
  16. P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form. Institute Math. Appl., Minneapolis, Preprint 593 (1989).  
  17. P.G. LeFloch, Graph solutions of nonlinear hyperbolic systems. J. Hyper. Differ. Equa.2 (2004) 643–689.  
  18. P.G. LeFloch and A.E. Tzavaras, Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal.30 (1999) 1309–1342.  
  19. R.J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comp. Phys.146 (1998) 346–365.  
  20. C. Parés, Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal.44 (2006) 300–321.  
  21. C. Parés and M. Castro, 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.  
  22. A.I. Volpert, The space BV and quasilinear equations. Math. USSR Sbornik73 (1967) 225–267.  

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