A numerical treatment of a superdegenerate equation with applications to the porous media equation
Authors:
M. Bertsch and R. Dal Passo
Journal:
Quart. Appl. Math. 48 (1990), 133-152
MSC:
Primary 65M12; Secondary 35K65, 76S05
DOI:
https://doi.org/10.1090/qam/1040238
MathSciNet review:
MR1040238
Full-text PDF Free Access
References |
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Additional Information
- D. G. Aronson, Nonlinear diffusion problems, Free boundary problems: theory and applications, Vol. I, II (Montecatini, 1981) Res. Notes in Math., vol. 78, Pitman, Boston, MA, 1983, pp. 135–149. MR 714914
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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), no. 4, 639–658. MR 704481, DOI https://doi.org/10.1137/0514049
- D. G. Aronson, L. A. Caffarelli, and Juan Luis Vázquez, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math. 38 (1985), no. 4, 375–404. MR 792397, DOI https://doi.org/10.1002/cpa.3160380404
V. F. Baklanovskaja, The numerical solution of a one-dimensional problem for equations of nonstationary filtrations, Zh. Vychisl. Mat. i Mat. Fiz. 1, 461–469 (1961)
A. E. Berger, H. Brézis, and J. C. W. Rogers, A numerical method for solving the problem ${u_t} - \Delta f\left ( u \right ) = 0$, RAIRO Anal. Num. 13, 297–312 (1979)
J. G. Berryman, Slow diffusion on the line, J. Math. Phys. 21, 1326–1331 (1980)
- Michiel Bertsch, Roberta Dal Passo, and Maura Ughi, Nonuniqueness of solutions of a degenerate parabolic equation, Ann. Mat. Pura Appl. (4) 161 (1992), 57–81. MR 1174811, DOI https://doi.org/10.1007/BF01759632
- M. Bertsch, C. J. van Duijn, J. R. Esteban, and Hong Fei Zhang, Regularity of the free boundary in a doubly degenerate parabolic equation, Comm. Partial Differential Equations 14 (1989), no. 3, 391–412. MR 987058, DOI https://doi.org/10.1080/03605308908820601
- Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101 (1979), no. 6, 1193–1218. MR 548877, DOI https://doi.org/10.2307/2374136
- Roberta Dal Passo and Stephan Luckhaus, A degenerate diffusion problem not in divergence form, J. Differential Equations 69 (1987), no. 1, 1–14. MR 897437, DOI https://doi.org/10.1016/0022-0396%2887%2990099-4
- E. DiBenedetto and David Hoff, An interface tracking algorithm for the porous medium equation, Trans. Amer. Math. Soc. 284 (1984), no. 2, 463–500. MR 743729, DOI https://doi.org/10.1090/S0002-9947-1984-0743729-3
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M. E. Rose, Numerical methods for a general class of porous medium equations, Argonne Nat. Lab. Rep., Argonne, Illinois (1980)
G. Rosen, Nonlinear heat conduction in solid H$_{2}$, Phys. Rev. B 19, 2398–2399 (1979)
E. A. Socolovsky, Difference schemes for degenerate nonlinear parabolic equations, preprint Carnegie-Mellon Univ., Pittsburgh (1983)
- Maura Ughi, A degenerate parabolic equation modelling the spread of an epidemic, Ann. Mat. Pura Appl. (4) 143 (1986), 385–400. MR 859613, DOI https://doi.org/10.1007/BF01769226
- Juan L. Vázquez, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), no. 2, 717–737. MR 752500, DOI https://doi.org/10.1090/S0002-9947-1984-0752500-8
D. G. Aronson, Nonlinear diffusion problems, Free Boundary Problems: Theory and Applications, Vol. 1 (eds. A. Fasano and M. Primicerio), Research Notes in Mathematics 78, Pitman, London, 1983, pp. 134–149
D. G. Aronson, The porous medium equation, Some Problems in Nonlinear Diffusion (eds. A. Fasano and M. Primicerio), Lecture Notes in Mathematics 1224 (CIME Foundation Series), Springer, Berlin, (1986)
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, 639–658 (1983)
D. G. Aronson, L. A. Caffarelli, and J. L. Vazquez, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math. 38, 375–404 (1985)
V. F. Baklanovskaja, The numerical solution of a one-dimensional problem for equations of nonstationary filtrations, Zh. Vychisl. Mat. i Mat. Fiz. 1, 461–469 (1961)
A. E. Berger, H. Brézis, and J. C. W. Rogers, A numerical method for solving the problem ${u_t} - \Delta f\left ( u \right ) = 0$, RAIRO Anal. Num. 13, 297–312 (1979)
J. G. Berryman, Slow diffusion on the line, J. Math. Phys. 21, 1326–1331 (1980)
M. Bertsch, R. Dal Passo, and M. Ughi, Nonuniqueness of solutions of a degenerate parabolic equation, preprint no. 4, Ist. Appl. Calc., Rome (1988)
M. Bertsch, C. J. Van Duyn, J. R. Esteban, and Zhang Hongfei, Regularity of the free boundary in a doubly degenerate parabolic equation, Comm. Partial Differential Equations 14, 391–412 (1989)
L. A. Caffarelli and A. Friedman, Regularity of the free boundary for the one-dimensional flow of gas in a porous medium, Amer. J. Math. 101, 1193–1281 (1979)
R. Dal Passo and S. Luckhaus, On a degenerate diffusion problem not in divergence form, to appear in J. Differential Equations 69, 1–14 (1987)
E. DiBenedetto and D. Hoff, An interface tracking algorithm for the porous medium equation, Trans. Amer. Math. Soc. 288, 463–500 (1984)
J. L. Graveleau and P. Jamet, A finite difference approach to some degenerate nonlinear parabolic equations, SIAM J. Appl. Math. 20, 199–223 (1971)
M. E. Gurtin, R. C. MacCamy, and E. A. Socolovsky, A coordinate transformation for the porous media equation that renders the free-boundary stationary, Quart. Appl. Math. 42, 345–357 (1984)
D. Hoff, A linearly implicit finite difference scheme for the one-dimensional porous medium equation, Math. Comp. 45, 23–33 (1985)
M. Mimura, T. Nakaki, and K. Tomoeda, A numerical approach to interface curves for some nonlinear diffusion equations, Japan J. Appl. Math. 1, 93–139 (1984)
M. Mimura and K. Tomoeda, Numerical approximations to interface curves for a porous media equation, Hiroshima Math. J. 13, 273–294 (1983)
M. E. Rose, Numerical methods for flows through porous media—I, Math. Comp. 40, 435–467 (1983)
M. E. Rose, Numerical methods for a porous medium equation, Argonne Nat. Lab. Rep., Argonne, Illinois (1978)
M. E. Rose, Numerical methods for a general class of porous medium equations, Argonne Nat. Lab. Rep., Argonne, Illinois (1980)
G. Rosen, Nonlinear heat conduction in solid H$_{2}$, Phys. Rev. B 19, 2398–2399 (1979)
E. A. Socolovsky, Difference schemes for degenerate nonlinear parabolic equations, preprint Carnegie-Mellon Univ., Pittsburgh (1983)
M. Ughi, A degenerate parabolic equation modelling the spread of an epidemic, Ann. Mat. Pura Appl. 143, 385–400 (1986)
J. L. Vazquez, The interface of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285, 717–737 (1984)
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Article copyright:
© Copyright 1990
American Mathematical Society