Coclite, Giuseppe Maria; Di Ruvo, Lorenzo; Ernest, Jan; Mishra, Siddhartha Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. (English) Zbl 1284.35276 Netw. Heterog. Media 8, No. 4, 969-984 (2013). Summary: 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. Cited in 19 Documents MSC: 35L65 Hyperbolic conservation laws 35L77 Higher-order quasilinear hyperbolic equations Keywords:conservation laws; discontinuous fluxes; capillarity approximation PDFBibTeX XMLCite \textit{G. M. Coclite} et al., Netw. Heterog. Media 8, No. 4, 969--984 (2013; Zbl 1284.35276) Full Text: DOI