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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