Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

Sébastien Martin; Julien Vovelle

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 699-727
  • ISSN: 0764-583X

Abstract

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This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).

How to cite

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Martin, Sébastien, and Vovelle, Julien. "Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 699-727. <http://eudml.org/doc/250348>.

@article{Martin2008,
abstract = { This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory). },
author = {Martin, Sébastien, Vovelle, Julien},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Finite Volume scheme; conservation law; discontinuous flux.; finite volume scheme; discontinuous flux; numerical examples; Cauchy-Dirichlet problem; convergence; entropy solution; Riemann problems},
language = {eng},
month = {7},
number = {5},
pages = {699-727},
publisher = {EDP Sciences},
title = {Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function},
url = {http://eudml.org/doc/250348},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Martin, Sébastien
AU - Vovelle, Julien
TI - Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 699
EP - 727
AB - This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).
LA - eng
KW - Finite Volume scheme; conservation law; discontinuous flux.; finite volume scheme; discontinuous flux; numerical examples; Cauchy-Dirichlet problem; convergence; entropy solution; Riemann problems
UR - http://eudml.org/doc/250348
ER -

References

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