Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An interface tracking algorithm for the porous medium equation


Authors: E. DiBenedetto and David Hoff
Journal: Trans. Amer. Math. Soc. 284 (1984), 463-500
MSC: Primary 65M10; Secondary 35K55, 35Q20, 35R35, 76S05
DOI: https://doi.org/10.1090/S0002-9947-1984-0743729-3
MathSciNet review: 743729
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation $ {u_t} = {({u^m})_{x\,x}},m > 1$.

The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution.

The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form $ \parallel {({u^{m - 1}})_{x\,x}}\;\parallel _{1,{\mathbf{R}}} \leqslant c/t$.

We prove that both the approximate solution and the approximate interfaces converge to the correct ones.

Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.


References [Enhancements On Off] (What's this?)

  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461-467. MR 0247303 (40:571)
  • [3] -, Regularity properties of flows through porous media: the interface, Arch. Rational Mech. Anal. 37 (1970), 1-10. MR 0255996 (41:656)
  • [4] -, Regularity properties of flows through porous media: a counterexample, SIAM J. Appl. Math. 19 (1970), 200-207. MR 0265774 (42:683)
  • [5] D. G. Aronson and Ph. Benilan, Régularité des solutions de l'équation des milieux poreux dans $ {{\mathbf{R}}^N}$, C. R. Acad. Sci. Paris, Vie Académique 288 (1979), 103-105. MR 524760 (82i:35090)
  • [6] G. I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mekh. 16 (1952), 67-78. MR 0046217 (13:700a)
  • [7] A. Fasano and M. Primicerio, Convergence of Hüber's method for heat conduction problems with change of phase, Z. Angew. Math. Mech. 53 (1973), 341-348. MR 0400930 (53:4760)
  • [8] B. H. Gilding, Hölder continuity of solutions of parabolic equations, J. London Math. Soc. (2) 13 (1976), 103-106. MR 0399658 (53:3501)
  • [9] J. L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equations, SIAM J. Appl. Math. 20 (1971), 199-223. MR 0290600 (44:7780)
  • [10] A. Hüber, Hauptaufsätze über das Fortschreiten der Schmelzgrenze in einem linearen Leiter, Z. Angew. Math. Mech. 19 (1939), 1-21.
  • [11] B. F. Knerr, The porous medium equation in one dimension, Trans. Amer. Math. Soc. 234 (1977), 381-415. MR 0492856 (58:11917)
  • [12] S. N. Kruškov, Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Mat. Zametki 6 (1969), 97-108. MR 0249832 (40:3073)
  • [13] M. Mimura and K. Tomoeda, Numerical approximations for interface curves to a porous media equation (to appear). MR 707183 (85c:65117)
  • [14] G. Minty, Monotone (non linear) operators in Hilbert space, Duke Math. J. 29 (1967), 341-346. MR 0169064 (29:6319)
  • [15] O. A. Oleinik, A. S. Kalashnikov and Chzou Yui-Lin, The Cauchy problem and boundary problems for equations of the type of unsteady filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667-704. MR 0099834 (20:6271)
  • [16] R. E. Pattle, Diffusion from an instantaneous point source with concentration dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409. MR 0114505 (22:5326)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 65M10, 35K55, 35Q20, 35R35, 76S05

Retrieve articles in all journals with MSC: 65M10, 35K55, 35Q20, 35R35, 76S05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0743729-3
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society