Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

Giuseppe Maria Coclite; Lorenzo di Ruvo; Jan Ernest; Siddhartha Mishra; Giuseppe Maria Coclite; Lorenzo di Ruvo; Jan Ernest; Siddhartha Mishra

doi:10.3934/nhm.2013.8.969

Networks and Heterogeneous Media

2013, Volume 8, Issue 4: 969-984. doi: 10.3934/nhm.2013.8.969

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Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

1.
Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari
2.
Department of Mathematics, University of Bari, via E. Orabona 4, 70125 Bari
3.
Seminar for Applied Mathematics (SAM), ETH Zürich, HG G 57.2, Rämistrasse 101, 8092 Zürich

Received: 01 October 2012 Revised: 01 April 2013
Primary: 35L65; Secondary: 35L77.

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.
- Conservation laws,
- discontinuous fluxes,
- capillarity approximation
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes[J]. Networks and Heterogeneous Media, 2013, 8(4): 969-984. doi: 10.3934/nhm.2013.8.969

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Abstract

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

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