Difference schemes for degenerate parabolic equations
Author:
E. A. Socolovsky
Journal:
Math. Comp. 47 (1986), 411420
MSC:
Primary 65M10
MathSciNet review:
856694
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [1]
Viorel
Barbu, Nonlinear semigroups and differential equations in Banach
spaces, Editura Academiei Republicii Socialiste România,
Bucharest, 1976. Translated from the Romanian. 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]
Alan
E. Berger, Haïm
Brézis, and Joel
C. W. Rogers, A numerical method for solving the problem
𝑢_{𝑡}Δ𝑓(𝑢)=0, RAIRO Anal.
Numér. 13 (1979), no. 4, 297–312
(English, with French summary). MR 555381
(81g:65120)
 [5]
H.
Brezis, M.
G. Crandall, and A.
Pazy, Perturbations of nonlinear maximal monotone sets in Banach
space, Comm. Pure Appl. Math. 23 (1970),
123–144. MR 0257805
(41 #2454)
 [6]
H.
Brézis and A.
Pazy, Convergence and approximation of semigroups of nonlinear
operators in Banach spaces, J. Functional Analysis 9
(1972), 63–74. MR 0293452
(45 #2529)
 [7]
Haïm
Brézis and Walter
A. Strauss, Semilinear secondorder elliptic equations in
𝐿¹, J. Math. Soc. Japan 25 (1973),
565–590. MR 0336050
(49 #826)
 [8]
M.
G. Crandall and T.
M. Liggett, Generation of semigroups of nonlinear transformations
on general Banach spaces, Amer. J. Math. 93 (1971),
265–298. MR 0287357
(44 #4563)
 [9]
M.
G. Crandall and A.
Pazy, Nonlinear evolution equations in Banach spaces, Israel
J. Math. 11 (1972), 57–94. MR 0300166
(45 #9214)
 [10]
Michael
G. Crandall, Semigroups of nonlinear transformations in Banach
spaces, Contributions to nonlinear functional analysis (Proc. Sympos.,
Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1971), Academic Press,
New York, 1971, pp. 157–179. Publ. Math. Res. Center Univ.
Wisconsin, No. 27. MR 0470787
(57 #10532)
 [11]
J.
Descloux, On the equation of Boussinesq, Topics in numerical
analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll., Dublin, 1976),
Academic Press, London, 1977, pp. 81–102. MR 0659075
(58 #31945)
 [12]
J.
Ildefonso Díaz Díaz, Solutions with compact support
for some degenerate parabolic problems, Nonlinear Anal.
3 (1979), no. 6, 831–847. MR 548955
(80i:35107), http://dx.doi.org/10.1016/0362546X(79)900518
 [13]
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
(85i:65119), http://dx.doi.org/10.1090/S00029947198407437293
 [14]
L. C. Evans, Nonlinear Evolution Equations, MRC, TSR No. 1568, 1975.
 [15]
Lawrence
C. Evans, Differentiability of a nonlinear semigroup in
𝐿¹, J. Math. Anal. Appl. 60 (1977),
no. 3, 703–715. MR 0454360
(56 #12611)
 [16]
David
Gilbarg and Neil
S. Trudinger, Elliptic partial differential equations of second
order, SpringerVerlag, Berlin, 1977. Grundlehren der Mathematischen
Wissenschaften, Vol. 224. MR 0473443
(57 #13109)
 [17]
Morton
E. Gurtin and Richard
C. MacCamy, On the diffusion of biological populations, Math.
Biosci. 33 (1977), no. 12, 35–49. MR 0682594
(58 #33147)
 [18]
Morton
E. Gurtin, Richard
C. MacCamy, and Eduardo
A. Socolovsky, A coordinate transformation for the porous media
equation that renders the free boundary stationary, Quart. Appl. Math.
42 (1984), no. 3, 345–357. MR 757173
(86d:35076)
 [19]
R.
C. MacCamy and Eduardo
Socolovsky, A numerical procedure for the porous media
equation, Comput. Math. Appl. 11 (1985),
no. 13, 315–319. Hyperbolic partial differential equations, II.
MR 787446
(86k:76064), http://dx.doi.org/10.1016/08981221(85)901567
 [20]
R.
H. Martin Jr., A global existence theorem for
autonomous differential equations in a Banach space, Proc. Amer. Math. Soc. 26 (1970), 307–314. MR 0264195
(41 #8791), http://dx.doi.org/10.1090/S00029939197002641956
 [21]
Masayasu
Mimura, Tatsuyuki
Nakaki, and Kenji
Tomoeda, A numerical approach to interface curves for some
nonlinear diffusion equations, Japan J. Appl. Math. 1
(1984), no. 1, 93–139. MR 839309
(87j:65111), http://dx.doi.org/10.1007/BF03167863
 [22]
Isao
Miyadera and Shinnosuke
Ôharu, Approximation of semigroups of nonlinear
operators, Tôhoku Math. J. (2) 22 (1970),
24–47. MR
0262874 (41 #7479)
 [23]
R.
E. Pattle, Diffusion from an instantaneous point source with a
concentrationdependent coefficient, Quart. J. Mech. Appl. Math.
12 (1959), 407–409. MR 0114505
(22 #5326)
 [24]
Michael
E. Rose, Numerical methods for flows through
porous media. I, Math. Comp.
40 (1983), no. 162, 435–467. MR 689465
(85a:65146), http://dx.doi.org/10.1090/S00255718198306894656
 [25]
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.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M10
Retrieve articles in all journals
with MSC:
65M10
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
