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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).
@article{M2AN_2008__42_5_699_0,
author = {Martin, S\'ebastien and Vovelle, Julien},
title = {Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {699--727},
publisher = {EDP-Sciences},
volume = {42},
number = {5},
year = {2008},
doi = {10.1051/m2an:2008023},
mrnumber = {2454620},
zbl = {1155.65071},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008023/}
}
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 PY - 2008 SP - 699 EP - 727 VL - 42 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008023/ DO - 10.1051/m2an:2008023 LA - en ID - M2AN_2008__42_5_699_0 ER -
%0 Journal Article %A Martin, Sébastien %A Vovelle, Julien %T Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2008 %P 699-727 %V 42 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an:2008023/ %R 10.1051/m2an:2008023 %G en %F M2AN_2008__42_5_699_0
Martin, Sébastien; Vovelle, Julien. Convergence of implicit finite volume methods for scalar conservation laws with discontinuous flux function. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 5, pp. 699-727. doi: 10.1051/m2an:2008023
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