Free Access
Volume 50, Number 5, September-October 2016
Page(s) 1289 - 1331
Published online 14 July 2016
  1. K. Aziz and A. Settari, Fundamentals of petroleum reservoir simulation. Applied Science Publishers, London (1979). [Google Scholar]
  2. C. Chalons and P.G. Lefloch, A fully-discrete scheme for diffusive-dispersive conservation laws. Numer. Math. 89 (2001) 493–509. [CrossRef] [MathSciNet] [Google Scholar]
  3. G.-Q. Chen, Compactness methods and nonlinear hyperbolic conservation laws. In some current topics on nonlinear conservation laws. AMS, Providence, RI (2000) 33–75. [Google Scholar]
  4. G.M. Coclite, L. di Ruvo, J. Ernest and S. Mishra, Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Netw. Heterog. Media. 8 (2013) 969–984. [Google Scholar]
  5. R.J. DiPerna, Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82 (1983) 27–70. [CrossRef] [Google Scholar]
  6. J. Ernest, P.G. Lefloch and S. Mishra, Schemes with Well-Controlled dissipation. I: Non-classical shock waves, SIAM J. Numer. Math. 53 (2015) 674–699. [CrossRef] [Google Scholar]
  7. S.M. Hassanizadeh and W.G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13 (1990) 169–186. [Google Scholar]
  8. B.T. Hayes and P.G. Lefloch, Nonclassical shock waves and kinetic relations. Strictly hyperbolic systems, Preprint no. 357. CMAP, Ecole Polytechnique, Palaiseau, France (1996). [Google Scholar]
  9. B.T. Hayes and P.G. Lefloch, Nonclassical shocks and kinetic relations: strictly hyperbolic systems. SIAM. J. Math. Anal. 31 (2000) 941–991. [CrossRef] [MathSciNet] [Google Scholar]
  10. T.Y. Hou and P.G. Lefloch, Why nonconservative schemes converge to wrong solutions. Error analysis. Math. Comput. 62 (1994) 497–530. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  11. S. Hwang and A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservations laws: Application to relaxation and diffusion-dispersion approximations. Commun. Partial Differ. Equ. 27 (2002) 1229–1254. [CrossRef] [Google Scholar]
  12. D. Jacobs, W.R. McKinney and M. Shearer, Traveling wave solutions of the modified Korteweg-deVries Burgers equation. J. Differ. Equ. 116 (1995) 448–467. [Google Scholar]
  13. K.H. Karlsen and J.D. Towers, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux, Chinese Ann. Math. Ser. B. 25 (2004) 287–318. [Google Scholar]
  14. K.H. Karlsen, N.H. Risebro and J.D. Towers, L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3 (2003) 1–49. [Google Scholar]
  15. K.H. Karlsen, S. Mishra and N.H. Risebro, Convergence of finite volume schemes for triangular systems of conservation laws. Numer. Math. 111 (2008) 559–589. [CrossRef] [Google Scholar]
  16. F. Kissling and C. Rohde, The computation of nonclassical shock waves with a heterogeneous multi-scale method. Netw. Heterog. Media. 5 (2010) 661–674. [CrossRef] [MathSciNet] [Google Scholar]
  17. F. Kissling and K.H. Karlsen, On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure. ZAMM, Z. Angew. Math. Mech. 94 (2014) 678–689. [CrossRef] [MathSciNet] [Google Scholar]
  18. C.I. Kondo and P.G. Lefloch, Zero diffusion-dispersion limits for scalar conservation laws. SIAM J. Math. Anal. 33 (2002) 1320–1329,. [CrossRef] [MathSciNet] [Google Scholar]
  19. S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S) 81 (1970) 228–255. [MathSciNet] [Google Scholar]
  20. P.G. Lefloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves. Lectures in Mathematics. Birkhäuser, Basel (2002). [Google Scholar]
  21. P.G. Lefloch and S. Mishra, Numerical methods with controlled dissipation for small-scale dependent shocks. Acta Numer. 23 (2014) 743–816. [CrossRef] [MathSciNet] [Google Scholar]
  22. R.J. Leveque, Numerical Methods for Conservation Laws. Birkhauser Verlag, Boston (1992). [Google Scholar]
  23. Y. Lu, Hyperbolic conservation laws and the compensated compactness method. Vol. 128 of Chapman and Hall/CRC Monographs and surveys in Pure and Applied Mathematics. Chapman and Hall/CRC, Boca Raton, FL (2003). [Google Scholar]
  24. F. Murat, Compacite par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5 (1978) 489–507. [MathSciNet] [Google Scholar]
  25. O.A. Oleĭnik, Convergence of certain difference schemes. Soviet Math. Dokl. 2 (1961) 313–316. [MathSciNet] [Google Scholar]
  26. B. Perthame and P.E. Souganidis, A limiting case for velocity averaging. Ann. Sci. E.N.S. 31 (1998) 591–598. [Google Scholar]
  27. M.E. Schonbek, Convergence of solution to nonlinear dispersive equations. Commun. Partial Differ. Equ. 7 (1982) 959–1000. [CrossRef] [MathSciNet] [Google Scholar]
  28. L. Tartar, Compensated compactness and applications to partial differential equations. Vol. 4 of Research Notes in Mathematics, Nonlinear Analysis and Mechanics. Heriot-Symposium 4 (1979) 136–212. [Google Scholar]
  29. J.D. Towers, Convergence of a finite difference scheme for conservation laws with a discontinous flux. SIAM. J. Numer. Anal. 38 (2000) 681–698. [CrossRef] [MathSciNet] [Google Scholar]
  30. C.J. van Duijn, L.A. Peletier and I.S. Pop, A new class of entropy solutions of the Buckley–Leverett equation. SIAM J. Math. Anal. 39 (2007) 507–536. [CrossRef] [MathSciNet] [Google Scholar]
  31. A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws. Arch. Rational Mech. Anal. 160 (2001) 181–193. [Google Scholar]
  32. A.I. Vol’pert, Generalized solutions of degenerate second-order quasilinear parabolic and elliptic equations. Adv. Differ. Equ. 5 (2000) 1493–1518. [Google Scholar]
  33. C.C. Wu, New theory of MHD shock waves, in Viscous Profiles and Numerical Methods for Shock Waves, edited by M. Shearer. SIAM, Philadelphia, PA (1991) 209–236. [Google Scholar]

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