A Finite Difference Approach to Some Degenerate Nonlinear Parabolic Equations
J. L. Graveleau, P. Jamet · SIAM Journal on Applied Mathematics · 1971
Previous article Next article A Finite Difference Approach to Some Degenerate Nonlinear Parabolic EquationsJ. L. Graveleau and P. JametJ. L. Graveleau and P. Jamethttps://doi.org/10.1137/0120027PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math., 17 (1969), 461–467 10.1137/0117045 MR0247303 0187.03401 LinkISIGoogle Scholar[2] D. G. Aronson, Regularity properties of flows through porous media: A counterexample., SIAM J. Appl. Math., 19 (1970), 299–307 10.1137/0119027 MR0265774 0255.76099 LinkISIGoogle Scholar[3] D. G. Aronson, Regularity properties of flows through porous media: The interface., Arch. Rational Mech. Anal., 37 (1970), 1–10 10.1007/BF00249496 MR0255996 0202.37901 CrossrefISIGoogle Scholar[4] Jean-Pierre Aubin, Un théorème de compacité, C. R. Acad. Sci. Paris, 256 (1963), 5042–5044 MR0152860 0195.13002 Google Scholar[5] Ju. A. Dubinskii˘, Weak convergence for nonlinear elliptic and parabolic equations, Mat. Sb. (N.S.), 67 (109) (1965), 609–642 MR0190546 Google Scholar[6] P. Jamet, , P. Lascaux and , P.-A. Raviart, Une méthode de résolution numérique des équations de Navier-Stokes, Numer. Math., 16 (1970), 93–114 10.1007/BF02308863 MR0280020 0203.48403 CrossrefISIGoogle Scholar[7] A. S. Kalashnikov, On the occurrence of singularities in the solution of the equation of non-stationary filtration, Zh. Vychisl. Mat. i Mat. Fiz., 7 (1967), 440–444 Google Scholar[8] S. N. Kružkov, The method of finite differences for a nonlinear equation of the first order with several independent variables, Z. Vyčisl. Mat. i Mat. Fiz., 6 (1966), 884–894 MR0203205 Google Scholar[9] P. Lascaux, Doctoral thesis, Paris, to appear Google Scholar[10] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969xx+554, Paris MR0259693 0189.40603 Google Scholar[11] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937 Google Scholar[12] W. F. Noh, A time-dependent, two-space dimensional, coupled Eulerian–Lagrange codeMethods in Computational Physics, Academic Press, New York, 1964, 11–107 Google Scholar[13] O. A. Olei˘nik, Discontinuous solutions of non-linear differential equations, Uspehi Mat. Nauk (N.S.), 12 (1957), 3–73 MR0094541 0131.31803 Google Scholar[14] O. A. Olei˘nik, , A. S. Kalašinkov and , Yui˘-Lin Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR. Ser. Mat., 22 (1958), 667–704 MR0099834 Google Scholar[15] P. A. Raviart, Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénérées, Arch. Rational Mech. Anal., 25 (1967), 64–80 10.1007/BF00281422 MR0215544 0153.42202 CrossrefISIGoogle Scholar[16] E. S. Sabinina, On a class of quasilinear parabolic equations, not solvable for the time derivative, Sibirsk. Mat. Ž., 6 (1965), 1074–1100 MR0190552 Google Scholar[17] E. S. Sabinina, On a class of non-linear degenerate parabolic equations, Dokl. Akad. Nauk SSSR, 143 (1962), 794–797 MR0132926 0122.33503 Google Scholar[18] S. L. Sobolev, Applications of functional analysis in mathematical physics, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963vii+239 MR0165337 0123.09003 CrossrefGoogle Scholar[19] Y. Zel'dovich and , Y. Raizer, Physics of Shock Waves and High-temperature Hydrodynamic Phenomena, Academic Press, New York, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails An asymptotic preserving scheme for a tumor growth model of porous medium type7 February 2022 | ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 56, No. 1 Cross Ref Finite volume element approximation for nonlinear diffusion problems with degenerate diffusion coefficientsApplied Numerical Mathematics, Vol. 140 Cross Ref Numerical methods for porous medium equation by an energetic variational approachJournal of Computational Physics, Vol. 385 Cross Ref An explicit finite-difference scheme for one-dimensional Generalized Porous Medium Equations: Interface tracking and the hole filling problem16 June 2016 | ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 50, No. 4 Cross Ref On the finite difference approximation for blow-up solutions of the porous medium equation with a sourceApplied Numerical Mathematics, Vol. 65 Cross Ref Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion LimitS. 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