Difference schemes for degenerate parabolic equations
Author:
E. A. Socolovsky
Journal:
Math. Comp. 47 (1986), 411420
MSC:
Primary 65M10
MathSciNet review:
856694
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Abstract: Diagonal dominant implicitdifference schemes approximating a porous media type class of multidimensional nonlinear equations are shown to generate semigroups in an approximate space, and the rate of convergence to the semigroup solution in is given. The numerical schemes proposed by Berger et al. in [4] are described and a proof of convergence for the fully discrete algorithms is outlined. Numerical experiments are discussed.
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E. A. Socolovsky, On Numerical Methods for Degenerate Parabolic Problems, Thesis, CarnegieMellon University, August, 1984.
 [1]
 V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. MR 0390843 (52:11666)
 [2]
 G. I. Barenblatt, "On certain nonstationary motions of liquids and gases in porous media," Prikl. Mat. Mekh., v. 16, 1952, pp. 6778.
 [3]
 Ph. Benilan, Équations d'Évolution dans un Espace de Banach quelconque et Applications, Thesis, Univ. Paris XI, Orsay, 1972.
 [4]
 A. E. Berger, H. Brezis & J. C. W. Rogers, "A numerical method for solving the problem ," RAIRO Numer. Anal., v. 13, 1979, pp. 297312. MR 555381 (81g:65120)
 [5]
 H. Brezis, M. Crandall & A. Pazy, "Perturbations of nonlinear maximal monotone sets in Banach spaces," Comm. Pure Appl. Math., v. 23, 1970, pp. 123144. MR 0257805 (41:2454)
 [6]
 H. Brezis & A. Pazy, "Convergence and approximation of semigroups of nonlinear operators in Banach spaces," J. Funct. Anal., v. 9, 1972, pp. 6374. MR 0293452 (45:2529)
 [7]
 H. Brezis & W. A. Strauss, "Semilinear second order elliptic equations in ," J. Math. Soc. Japan, v. 25, 1973, pp. 565590. MR 0336050 (49:826)
 [8]
 M. Crandall & T. Liggett, "Generation of semigroups of nonlinear transformations on general Banach spaces," Amer. J. Math., v. 93, 1971, pp. 265298. MR 0287357 (44:4563)
 [9]
 M. Crandall & A. Pazy, "Nonlinear evolution equations in Banach spaces," Israel J. Math., v. 11, 1972, pp. 5794. MR 0300166 (45:9214)
 [10]
 M. Crandall, "Semigroups of nonlinear transformations in Banach spaces," in Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, ed.), Academic Press, New York, 1971, pp. 157179. MR 0470787 (57:10532)
 [11]
 J. Descloux, "On the equation of Boussinesq," in Topics in Numerical Analysis (J. J. H. Miller, ed.), Vol. 3, Academic Press, London, 1977, pp. 81102. MR 0659075 (58:31945)
 [12]
 J. I. Diaz Diaz, "Solutions with compact support for some degenerate parabolic problems," Nonlinear Anal., v. 3, 1979, pp. 831847. MR 548955 (80i:35107)
 [13]
 E. Dibenedetto & D. C. Hoff, "An interface tracking algorithm for the porous medium equation," Trans. Amer. Math. Soc., v. 284, 1984, pp. 463500. MR 743729 (85i:65119)
 [14]
 L. C. Evans, Nonlinear Evolution Equations, MRC, TSR No. 1568, 1975.
 [15]
 L. C. Evans, "Differentiability of a nonlinear semigroup in ," J. Math. Anal. Appl., v. 60, 1977, pp. 703715. MR 0454360 (56:12611)
 [16]
 D. Gilbarg & N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, SpringerVerlag, New York, 1977. MR 0473443 (57:13109)
 [17]
 M. E. Gurtin & R. C. MacCamy, "On the diffusion of biological populations," Math. Biosci., v. 33, 1977, pp. 3549. MR 0682594 (58:33147)
 [18]
 M. E. Gurtin, R. C. MacCamy & E. A. Socolovsky, "A coordinate transformation for the porous media equation that renders the freeboundary stationary," Quart. Appl. Math., v. 42, 1984, pp. 345357. MR 757173 (86d:35076)
 [19]
 R. C. MacCamy & E. A. Socolovsky, "A numerical procedure for the porous media equation," Comput. Math. Appl., v. 11, 1985, pp. 315319. MR 787446 (86k:76064)
 [20]
 R. H. Martin, Jr., "A global existence theorem for autonomous differential equations in a Banach space," Proc. Amer. Math. Soc., v. 26, 1970, pp. 307314. MR 0264195 (41:8791)
 [21]
 M. Mimura, R. Nakaki & K. Tomoeda, "A numerical approach to interface curves for some nonlinear diffusion equations," Japan J. Appl. Math., v. 1, 1984, pp. 93139. MR 839309 (87j:65111)
 [22]
 I. Miyadera & S. Oharu, "Approximation of semigroups of nonlinear operators," Tôhoku Math. J., v. 22, 1970, pp. 2447. MR 0262874 (41:7479)
 [23]
 R. C. Pattle, "Diffusion from an instantaneous point source with a concentrationdependent coefficient," Quart. J. Mech. Appl. Math., v. 12, 1959, pp. 407409. MR 0114505 (22:5326)
 [24]
 M. E. Rose, "Numerical methods for flows through porous media. I," Math. Comp., v. 40, 1983, pp. 435467. MR 689465 (85a:65146)
 [25]
 E. A. Socolovsky, On Numerical Methods for Degenerate Parabolic Problems, Thesis, CarnegieMellon University, August, 1984.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608566940
PII:
S 00255718(1986)08566940
Keywords:
Difference schemes,
degenerate nonlinear parabolic equations,
nonlinear semigroups
Article copyright:
© Copyright 1986
American Mathematical Society
