Abstract
Nonlinear diffusion equations in one dimensional spacev t =(v m) xx +vF(v) (m>1) appear in the fields of fluid dynamics, combustion theory, plasma physics and population dynamics. The most interesting phenomenon is the finite speed of propagation. Specifically, if the initial function has compact support, the solution has also compact support for later times, and there appears an interface betweenv>0 andv=0. The aim of this paper is to propose a finite difference scheme possessing the property that numerical solutions as well as numerical interfaces converge to the exact ones. Numerical examples are also presented.
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D. G. Aronson, L. A. Caffarelli and S. Kamin, How an initially stationary interface begins to move in porous medium flow, SIAM J. Math. Anal.,14 (1983), 639–658.
V. F. Baklanovskaya, The numerical solution of a one-dimensional problem for equations of nonstationary filtrations. Ž. Vyčisl. Mat. i Mat. Fiz.,1 (1961), 461–469.
G. I. Barenblatt, On some unsteady motions of a liquid and a gas in a porous medium, Prinkl. Mat. Meh.,16 (1952), 67–78.
H. Berestycki, B. Nicolaenko and B. Scheurer. Traveling wave solutions to reaction diffusion systems modeling combustion, Contemporary Math.17, 1983, 189–208.
J. G. Berryman and C. J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal.,74 (1980), 379–388.
M. Bertsch, R. Kersner and L. A. Peletier, Positivity versus localization in degenerate diffusion equations, to appear.
H. Brézis and M. G. Crandall, Uniqueness of solutions fo the initial-value problem foru t −δϕ(u)=0, J. Math. Pures Appl.,58 (1979), 153–163.
J. Buckmaster, Viscous sheets advancing over dry beds, J. Fluid Mech.,81 (1977), 735–756.
E. A. Carl, Population control in arctic ground squirrels, Ecology,52 (1971), 395–413.
E. DiBenedetto and D. Hoff, An interface tracking algorithm for the porous medium equation, to appear in Trans. Amer. Math. Soc.
J. L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equations, SIAM J. Appl. Math.,20 (1971), 199–223.
M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci.,33 (1977), 35–49.
D. Hoff, A linearly implicit finite difference scheme for the one-dimensional porous medium equation, preprint.
A. S. Kalashnikov, The propagation of disturbances in problems of non-linear heat conduction with absorption, Ž, Vyčisl. Mat. i Mat. Fiz.,14 (1974), 891–905.
B. F. Knerr, The behaviour of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension. Trans. Amer. Math. Soc.,249 (1979), 409–424.
C. J. Krebs and J. H. Myers. Population cycles in small mammals, Adv. in Ecol. Res.,8 (1974), 267–399.
E. W. Larsen and G. C. Pomraning, Asymptotic analysis of nonlinear Marshak waves, SIAM J. Appl. Math.,39 (1980), 201–212.
W. I. Newman, Some exact solutions to non-linear diffusion problem in population genetics and combustion, J. Theoret. Biol.,85 (1980), 325–334.
W. I. Newman and C. Sagan, Galactic civilizations: Population dynamics and interstellar diffusion, preprint.
R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math.,12 (1959), 407–409.
P. Y. Polubarinova-Kochina, Theory of Ground Water Movement. Princeton Univ. Press, 1962.
B. Quinn, Solutions with shocks, an example of anL 1-contractive semigroup. Comm. Pure Appl. Math.,24 (1971), 125–132.
P. Rosenau and S. Kamin, Thermal waves in an absorbing and convecting medium, Physica,8D (1983), 273–283.
A. E. Scheidegger, The Physics of Flow through Porous Media, Third edition, University of Toronto Press, 1974.
K. Tomoeda and M. Mimura, Numerical approximations to interface curves for a porous media equation, Hiroshima Math. J.,13 (1983), 273–294.
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Mimura, M., Nakaki, T. & Tomoeda, K. A numerical approach to interface curves for some nonlinear diffusion equations. Japan J. Appl. Math. 1, 93–139 (1984). https://doi.org/10.1007/BF03167863
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DOI: https://doi.org/10.1007/BF03167863