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December  2013, 8(4): 969-984. doi: 10.3934/nhm.2013.8.969

## 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, Italy 3 Seminar for Applied Mathematics (SAM), ETH Zürich, HG G 57.2, Rämistrasse 101, 8092 Zürich, Switzerland, Switzerland

Received  October 2012 Revised  April 2013 Published  November 2013

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.
Citation: Giuseppe Maria Coclite, Lorenzo di Ruvo, Jan Ernest, Siddhartha Mishra. Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes. Networks & Heterogeneous Media, 2013, 8 (4) : 969-984. doi: 10.3934/nhm.2013.8.969
##### References:
 [1] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions,, J. Hyp. Diff. Eqns., 2 (2005), 783.  doi: 10.1142/S0219891605000622.  Google Scholar [2] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux,, Arch. Ration. Mech. Anal., 201 (2011), 27.  doi: 10.1007/s00205-010-0389-4.  Google Scholar [3] E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253.  doi: 10.1017/S0308210500003863.  Google Scholar [4] K. Aziz and A. Settari, Fundamentals of Petroleum Reservoir Simulation,, Applied Science Publishers, (1979).   Google Scholar [5] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units,, Numer. Math., 97 (2004), 25.  doi: 10.1007/s00211-003-0503-8.  Google Scholar [6] G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation,, Comm. Partial Differential Equations, 31 (2006), 1253.  doi: 10.1080/03605300600781600.  Google Scholar [7] G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws,, Boll. Unione Mat. Ital. (9), 2 (2009), 275.   Google Scholar [8] C. Dafermos, Hyperbolic Conservation laws in Continuum Physics,, $3^{rd}$ edition, (2005).   Google Scholar [9] E. vanDuijn, L. A. Peletier and S. Pop, A new class of entropy solutions of the Buckley-Leverett equation,, SIAM J. Math. Anal., 39 (2007), 507.  doi: 10.1137/05064518X.  Google Scholar [10] T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function,, SIAM J. Math. Anal., 23 (1992), 635.  doi: 10.1137/0523032.  Google Scholar [11] R. Helmig, A. Weiss and B. I. Wohlmuth, Dynamic capillary effects in heterogeneous porous media,, Comp. Geosci., 11 (2007), 261.  doi: 10.1007/s10596-007-9050-1.  Google Scholar [12] S. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries,, Adv. Wat. Res., 13 (1990), 169.  doi: 10.1016/0309-1708(90)90040-B.  Google Scholar [13] H. Holden, K. H. Karlsen and D. Mitrovic, Zero diffusion-dispersion-smoothing limits for scalar conservation law with discontinuous flux function,, International Journal of Differential Equations, 2009 (2009), 1.   Google Scholar [14] H. Holden, K. H. Karlsen, D. Mitrovic and E. Y. Panov, Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function,, Acta Mathematica Scientia, 29B (2009), 573.  doi: 10.1016/S0252-9602(10)60004-5.  Google Scholar [15] K. H. Karlsen and F. Kissling, On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure,, Z. Angew. Math. Mech., ().   Google Scholar [16] K. H. Karlsen, N. H. Risebro and J. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients,, Skr. K. Nor. Vidensk. Selsk., 3 (2003), 1.   Google Scholar [17] F. Kissling and C. Rohde, The computation of nonclassical shock waves with a heterogeneous multiscale method,, Netw. Heterog. Media, 5 (2010), 661.  doi: 10.3934/nhm.2010.5.661.  Google Scholar [18] P. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory Of Classical and Non-Classical Shock Waves,, Lecture notes in Mathematics., (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar [19] S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media,, Comp. GeoSci., 14 (2010), 105.  doi: 10.1007/s10596-009-9135-0.  Google Scholar [20] F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl. (9), 60 (1981), 309.   Google Scholar [21] S. Mochon, An analysis of the traffic on highways with changing surface conditions,, Math. Model., 9 (1987), 1.  doi: 10.1016/0270-0255(87)90068-6.  Google Scholar [22] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Ration. Mech. Anal., 195 (2010), 643.  doi: 10.1007/s00205-009-0217-x.  Google Scholar [23] E. Yu. Panov, Erratum to: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Ration. Mech. Anal., 196 (2010), 1077.  doi: 10.1007/s00205-010-0303-0.  Google Scholar

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##### References:
 [1] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions,, J. Hyp. Diff. Eqns., 2 (2005), 783.  doi: 10.1142/S0219891605000622.  Google Scholar [2] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux,, Arch. Ration. Mech. Anal., 201 (2011), 27.  doi: 10.1007/s00205-010-0389-4.  Google Scholar [3] E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253.  doi: 10.1017/S0308210500003863.  Google Scholar [4] K. Aziz and A. Settari, Fundamentals of Petroleum Reservoir Simulation,, Applied Science Publishers, (1979).   Google Scholar [5] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units,, Numer. Math., 97 (2004), 25.  doi: 10.1007/s00211-003-0503-8.  Google Scholar [6] G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation,, Comm. Partial Differential Equations, 31 (2006), 1253.  doi: 10.1080/03605300600781600.  Google Scholar [7] G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws,, Boll. Unione Mat. Ital. (9), 2 (2009), 275.   Google Scholar [8] C. Dafermos, Hyperbolic Conservation laws in Continuum Physics,, $3^{rd}$ edition, (2005).   Google Scholar [9] E. vanDuijn, L. A. Peletier and S. Pop, A new class of entropy solutions of the Buckley-Leverett equation,, SIAM J. Math. Anal., 39 (2007), 507.  doi: 10.1137/05064518X.  Google Scholar [10] T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function,, SIAM J. Math. Anal., 23 (1992), 635.  doi: 10.1137/0523032.  Google Scholar [11] R. Helmig, A. Weiss and B. I. Wohlmuth, Dynamic capillary effects in heterogeneous porous media,, Comp. Geosci., 11 (2007), 261.  doi: 10.1007/s10596-007-9050-1.  Google Scholar [12] S. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries,, Adv. Wat. Res., 13 (1990), 169.  doi: 10.1016/0309-1708(90)90040-B.  Google Scholar [13] H. Holden, K. H. Karlsen and D. Mitrovic, Zero diffusion-dispersion-smoothing limits for scalar conservation law with discontinuous flux function,, International Journal of Differential Equations, 2009 (2009), 1.   Google Scholar [14] H. Holden, K. H. Karlsen, D. Mitrovic and E. Y. Panov, Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function,, Acta Mathematica Scientia, 29B (2009), 573.  doi: 10.1016/S0252-9602(10)60004-5.  Google Scholar [15] K. H. Karlsen and F. Kissling, On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure,, Z. Angew. Math. Mech., ().   Google Scholar [16] K. H. Karlsen, N. H. Risebro and J. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients,, Skr. K. Nor. Vidensk. Selsk., 3 (2003), 1.   Google Scholar [17] F. Kissling and C. Rohde, The computation of nonclassical shock waves with a heterogeneous multiscale method,, Netw. Heterog. Media, 5 (2010), 661.  doi: 10.3934/nhm.2010.5.661.  Google Scholar [18] P. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory Of Classical and Non-Classical Shock Waves,, Lecture notes in Mathematics., (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar [19] S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media,, Comp. GeoSci., 14 (2010), 105.  doi: 10.1007/s10596-009-9135-0.  Google Scholar [20] F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl. (9), 60 (1981), 309.   Google Scholar [21] S. Mochon, An analysis of the traffic on highways with changing surface conditions,, Math. Model., 9 (1987), 1.  doi: 10.1016/0270-0255(87)90068-6.  Google Scholar [22] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Ration. Mech. Anal., 195 (2010), 643.  doi: 10.1007/s00205-009-0217-x.  Google Scholar [23] E. Yu. Panov, Erratum to: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Ration. Mech. Anal., 196 (2010), 1077.  doi: 10.1007/s00205-010-0303-0.  Google Scholar
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