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

by G. M. Coclite and L. Di Ruvo and J. Ernest and S. Mishra

(Report number 2012-30)

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.

Keywords:

@Techreport{CDEM12_473,
  author = {G. M. Coclite and L. Di Ruvo and J. Ernest and S. Mishra},
  title = {Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2012-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2012/2012-30.pdf },
  year = {2012}
}

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