A linearly implicit finite-difference scheme for the one-dimensional porous medium equation
Author:
David Hoff
Journal:
Math. Comp. 45 (1985), 23-33
MSC:
Primary 65M15; Secondary 76S05
DOI:
https://doi.org/10.1090/S0025-5718-1985-0790642-6
MathSciNet review:
790642
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We present and analyze a linearly implicit finite-difference scheme for computing approximate solutions and interface curves for the porous medium equation in one space variable. Our scheme requires only that linear, tridiagonal systems of equations be solved at each time step. We derive error bounds for the approximate interface curves as well as for the approximate solutions under the rather mild mesh condition .
- [1] D. G. Aronson, "Regularity properties of flows through porous media," SIAM J. Appl. Math., v. 17, 1969, pp. 461-467. MR 0247303 (40:571)
- [2] D. G. Aronson, "Regularity properties of flows through porous media: the interface," Arch. Rational Mech. Anal., v. 37, 1970, pp. 1-10. MR 0255996 (41:656)
- [3] D. G. Aronson, "Regularity properties of flows through porous media: a counterexample," SIAM J. Appl. Math., v. 19, 1970, pp. 200-307. MR 0265774 (42:683)
- [4]
D. G. Aronson & Ph. Benilan, "Régularité des solutions de l'équation des milieux poreus dans
," C. R. Acad. Sci. Paris Vie Académique, v. 288, 1979, pp. 103-105. MR 524760 (82i:35090)
- [5] E. DiBenedetto & David Hoff, "An interface tracking algorithm for the porous medium equation," Trans. Amer. Math. Soc., v. 288, 1984, pp. 463-500. MR 743729 (85i:65119)
- [6] J. L. Graveleau & P. Jamet, "A finite difference approach to some degenerate nonlinear parabolic equations," SIAM J. Appl. Math., v. 20, 1971, 199-223. MR 0290600 (44:7780)
- [7] S. N. Kružkov, "Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications," Math. Notes, v. 6, no. 1, 1969, pp. 517-523.
- [8] M. Mimura & K. Tomoeda, "Numerical approximations for interface curves to a porous media equation." (To appear.) MR 737598 (85e:76054)
- [9] O. A. Oleĭnik, A. S. Kalashnikov & Chzou Yui-Lin, "The Cauchy problem and boundary problems for equations of the type of unsteady filtration," Izv. Akad. Nauk SSSR Ser. Mat., v. 22, 1958, pp. 667-704. (Russian) MR 0099834 (20:6271)
- [10] M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937.
Retrieve articles in Mathematics of Computation with MSC: 65M15, 76S05
Retrieve articles in all journals with MSC: 65M15, 76S05
Additional Information
DOI:
https://doi.org/10.1090/S0025-5718-1985-0790642-6
Article copyright:
© Copyright 1985
American Mathematical Society