An explicit finite-difference scheme for one-dimensional Generalized Porous Medium Equations: Interface tracking and the hole filling problem

Monsaingeon, Léonard

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 4, pp. 1011-1033.

Based on results of E. DiBenedetto and D. Hoff we propose an explicit finite-difference scheme for one dimensional Generalized Porous Medium Equations ${\partial }_{t}u={\partial }_{xx}^2\varphi \left(u\right)$. The scheme allows to track the moving free boundaries, and captures the so-called hole filling phenomenon when free boundaries collide. We prove the convergence of the discrete solution when the mesh parameter $\Delta x\to 0$. Finally, we provide numerical evidence of the convergence of the scheme.

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. Clarendon Press, Oxford, New York (2000). | MR | Zbl

D.G. Aronson, Regularity properties of flows through porous media: The interface. Arch. Rational Mech. Anal. 37 (1970) 1–10. | DOI | MR | Zbl

D.G. Aronson, Regularity propeties of flows through porous media. SIAM J. Appl. Math. 17 (1969) 461–467. | DOI | MR | Zbl

D.G. Aronson and P. Bénilan, Régularité des solutions de l’équation des milieux poreux dans ${𝐑}^N$. C. R. Acad. Sci. Paris Sér. A-B 288 (1979) A103–A105. | MR | Zbl

L.A. Caffarelli, J.L. Vázquez, and N.I. Wolanski, Lipschitz continuity of solutions and interfaces of the $N$-dimensional porous medium equation. Indiana Univ. Math. J. 36 (1987) 373–401. | DOI | MR | Zbl

L.A. Caffarelli and N. Wolanski, $C^{1,\alpha }$ regularity of the free boundary for the $N$-dimensional porous media equation. Commun. Pure Appl. Math. 43 (1990) 885–902. | DOI | MR | Zbl

M.G. Crandall and M. Pierre, Regularizing effects for $u_t=\Delta \varphi \left(u\right)$. Trans. Amer. Math. Soc. 274 (1982) 159–168. | MR | Zbl

B.E.J. Dahlberg and C.E. Kenig, Nonnegative solutions of generalized porous medium equations. Rev. Mat. Iberoamericana 2 (1986) 267–305. | DOI | MR | Zbl

P. Daskalopoulos and C.E. Kenig, Degenerate diffusions. Initial value problems and local regularity theory. Vol. 1 of EMS Tracts Math. European Mathematical Society (EMS), Zürich (2007). | MR | Zbl

P. Daskalopoulos and E. Rhee, Free-boundary regularity for generalized porous medium equations. Commun. Pure Appl. Anal. 2 (2003) 481–494. | DOI | MR | Zbl

A. De Pablo and J.L. Vázquez, Regularity of solutions and interfaces of a generalized porous medium equation in ${𝐑}^N$. Ann. Mat. Pura Appl. 158 (74) 51–74. | MR | Zbl

F. del Teso and J.L. Vázquezn, Finite difference method for a general fractional porous medium equation. Preprint arXiv:1307.2474 (2013).

E. DiBenedetto, Degenerate parabolic equations. Universitext. Springer-Verlag, New York (1993). | MR | Zbl

E. Dibenedetto and D. Hoff, An interface tracking algorithm for the porous medium equation. Trans. Amer. Math. Soc. 284 (1984) 463–500. | DOI | MR | Zbl

B.H. Gilding, Hölder continuity of solutions of parabolic equations. J. London Math. Soc. 13 (1976) 103–106. | DOI | MR | Zbl

J.L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equations. SIAM J. Appl. Math. 20 (223) 199–223. | MR | Zbl

D. Hoff, A linearly implicit finite-difference scheme for the one-dimensional porous medium equation. Math. Comput. 45 (1985) 23–33. | DOI | MR | Zbl

T. Nakaki and K. Tomoeda, A finite difference scheme for some nonlinear diffusion equations in an absorbing medium: support splitting phenomena. SIAM J. Numer. Anal. 40 (2002) 945–964. | DOI | MR | Zbl

P.E. Sacks, Continuity of solutions of a singular parabolic equation. Nonlin. Anal. 7 (1983) 387–409. | DOI | MR | Zbl

K. Tomoeda and M. Mimura, Numerical approximations to interface curves for a porous media equation. Hiroshima Math. J. 13 (1983) 273–294. | DOI | MR | Zbl

K. Tomoeda, Numerically repeated support splitting and merging phenomena in a porous media equation with strong absorption. J. Math-Industry 3 (2011) 61–68. | MR | Zbl

J.L. Vázquez, The porous medium equation. Oxford Math. Monogr. The Clarendon Press, Oxford University Press, Oxford (2007). Mathematical theory. | MR | Zbl

Q. Zhang and Z. Wu, Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method. J. Sci. Comput. 38 (2009) 127–148. | DOI | MR | Zbl