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CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION

R. Eymard1, C. Guichard2, Xavier Lhébrard3   

  1. 1. Université Gustave Eiffel, Laboratoire d'Analyse et de Mathématiques Appliquées (UMR 8050), UGE, UPEC, CNRS, F-77454, Marne-la-Vallée, France;
    2. Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, équipe ANGE, F-75005 Paris;
    3. Ecole Normale Supérieure de Rennes, France
  • Received:2018-12-23 Revised:2018-12-23 Published:2021-04-12
  • Contact: R. Eymard,Email:robert.eymard@u-pem.fr
  • Supported by:
    The authors thank Professor Thierry Gallouët for fruitful discussions. This work was supported by the French Agence Nationale de la Recherche (CHARMS project, ANR-16-CE06-0009).

R. Eymard, C. Guichard, Xavier Lhébrard. CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION[J]. Journal of Computational Mathematics, 2021, 39(3): 428-452.

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of θ for stabilising the scheme.

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[1] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183:3(1983), 311-341.
[2] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158:2(2004), 227-260.
[3] G. Amiez and P.A. Gremaud, On a numerical approach to Stefan-like problems, Numer. Math., 59:1(1991), 71-89.
[4] B. Andreianov, M. Bendahmane, K.H. Karlsen, and S. Ouaro, Well-posedness results for triply nonlinear degenerate parabolic equations, J. Differential Equations, 247:1(2009), 277-302.
[5] L. Beaude, K. Brenner, S. Lopez, R. Masson, and F. Smai, Non-isothermal compositional twophase Darcy flow:formulation and outflow boundary condition, In Finite volumes for complex applications VIII-hyperbolic, elliptic and parabolic problems, Springer Proc. Math. Stat., Springer, Cham, 200(2017), 317-325.
[6] M. Bertsch, P. de Mottoni, and L. A. Peletier, The Stefan problem with heating:appearance and disappearance of a mushy region, Trans. Amer. Math. Soc., 293:2(1986), 677-691.
[7] C. Cancès and C. Guichard, Numerical analysis of a robust free energy diminishing finite volume scheme for parabolic equations with gradient structure, Found. Comput. Math., 17:6(2017), 1525-1584.
[8] J. Carrillo, Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., 147:4(1999), 269-361.
[9] Z. Chen, Expanded mixed finite element methods for quasilinear second order elliptic problems, II, RAIRO Modél. Math. Anal. Numér., 32:4(1998), 501-520.
[10] D. De Vries, Thermal properties of soils, In W. R. van Wijk, editor, Physics of Plant Environment (W. R. van Wijk ed.), North-Holland Publishing Co., Amsterdam, 1964, pages 210-235.
[11] R.J. DiPerna and P.L. Lions, On the Cauchy problem for Boltzmann equations:global existence and weak stability, Ann. of Math., 130:2(1989), 321-366.
[12] J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, The gradient discretisation method, volume 82 of Mathematics & Applications, Springer International Publishing, 2018, https://hal.archives-ouvertes.fr/hal-01382358.
[13] C.M. Elliott, On the finite element approximation of an elliptic variational inequality arising from an implicit time discretization of the Stefan problem, IMA J. Numer. Anal., 1:1(1981), 115-125.
[14] C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J. Numer. Anal., 7:1(1987), 61-71.
[15] L.C. Evans, Partial differential equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010.
[16] R. Eymard, P. Feron, T. Gallouët, R. Herbin, and C. Guichard, Gradient schemes for the Stefan problem, International Journal on Finite Volumes, 2013, Volume 10 special.
[17] R. Eymard, T. Gallouët, C. Guichard, R. Herbin, and R. Masson, TP or not TP, that is the question, Comput. Geosci., 18:3-4(2014), 285-296.
[18] R. Eymard, C. Guichard, and R. Herbin, Small-stencil 3d schemes for diffusive flows in porous media, M2AN Math. Model. Numer. Anal., 46(2012), 265-290.
[19] R. Eymard, R. Herbin, and A. Michel, Mathematical study of a petroleum-engineering scheme, M2AN Math. Model. Numer. Anal., 37:6(2003), 937-972.
[20] T. Gallouët and R. Herbin, Mesure, Intégration, Probabilités, Références science. Ellipses, Paris, 2013.
[21] R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, In Finite volumes for complex applications V, ISTE, London, 2008, pages 659-692.
[22] I.C. Kim and N. Požár, Viscosity solutions for the two-phase Stefan problem, Comm. Partial Differential Equations, 36:1(2011), 42-66.
[23] F.A. Milner, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44:170(1985), 303-320.
[24] R. Nochetto and C. Verdi, Approximation of degenerate parabolic problems using numerical integration, SIAM Journal on Numerical Analysis, 25:4(1988), 784-814.
[25] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15:fasc. 1(1965), 189-258.
[26] F. Yi and T.M. Shih, Stefan problem with convection, Appl. Math. Comput., 95:2-3(1998), 139-154.
[1] Hua-zhong Tang,Gerald Warnecke. HIGH RESOLUTION SCHEMES FOR CONSERVATION LAWS AND CONVECTION-DIFFUSION EQUATIONS WITH VARYING TIME AND SPACE GRIDS [J]. Journal of Computational Mathematics, 2006, 24(2): 121-140.
[2] C. Clavero,J.J.H. Miller,E. O’Riordan,G.I. Shishkin . AN ACCURATE NUMERICAL SOLUTION OF A TWO DIMENSIONAL HEAT TRANSFERPROBLEM WITH A PARABOLIC BOUNDARY LAYER [J]. Journal of Computational Mathematics, 1998, 16(1): 27-039.
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