×

A numerical method for solving the problem \(u_t-\Delta f(u)=0\). (English) Zbl 0426.65052


MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
47H20 Semigroups of nonlinear operators
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] 1. D. G. ARONSON, Regularity Properties of Flows through Porous Media, S.I.A.M., J. Appl. Math., Vol. 17, 1969, pp. 461-467. Zbl0187.03401 MR247303 · Zbl 0187.03401
[2] 2. D. R. ATTHEY, A finite Difference Scheme for Melting Problems, J. Inst. Math. Appl., Vol. 13, 1974, pp. 353-366. MR351295
[3] 3. J. R. BAILLON and B. MERCIER, Convergence of Approximations to Nonlinear Semigroups, École Polytechnique Internal Report Number 24, December 1977.
[4] 4. Ph. BENILAN, Equations d’évolution dans un espace de Banach quelconque et applications, Thèse, Publications Mathématiques d’Orsay, n^\circ 25, 1972.
[5] 5. A. E. BERGER, The Truncation Method for the Solution of a Class of Variational Inequalities, R.A.I.R.O., Analyse Numérique, Vol. 10, 1976, pp. 29-42. Zbl0334.49012 MR455489 · Zbl 0334.49012
[6] 6. A. E. BERGER, M. CIMENT and J. C. W. ROGERS, Numerical Solution of a Diffusion Consumption Problem with a Free Boundary, S.I.A.M. J. Numer. Anal., Vol. 12, 1975, pp. 646-672. Zbl0317.65032 MR383779 · Zbl 0317.65032
[7] 7. A. E. BERGER, M. CIMENT and J. C. W. ROGERS, Numerical Solution of a Stefan Problem by a Technique of Alternating Phase Truncation, Séminaires IRIA, Analyse et contrôle des systèmes, Rocquencourt, Institut de Recherche d’Informatique et d’automatique, 1976, pp. 21-34.
[8] 8. A. E. BERGER and R. S. FALK, An Error Estimate for the Truncation Method for the Solution of Parabolic Obstacle Variational Inequalities, Math. Comp., Vol. 31, 1977 pp. 619-628. Zbl0367.65056 MR438707 · Zbl 0367.65056
[9] 9. H. BREZIS, On Some Degenerate Nonlinear Parabolic Equations, Nonlinear Functional Analysis, F. BROWDER, Ed., Proc. Symp. in pure math., Vol. 18, A.M.S., 1970, pp. 28-38. Zbl0231.47034 MR273468 · Zbl 0231.47034
[10] 10. H. BREZIS and A. PAZY, Convergence and Approximation of Semigroups of Nonlinear Operators in Banach Spaces, J. Func. Anal., Vol. 9, 1972, pp. 63-74. Zbl0231.47036 MR293452 · Zbl 0231.47036
[11] 11. H. BREZIS and W. A. STRAUSS, Semi-linear second-order elliptic equations in L 1 , J. Math. Soc. Japan, Vol. 25, 1973, pp. 565-590. Zbl0278.35041 MR336050 · Zbl 0278.35041
[12] 12. M. G. CRANDALL, An Introduction to Evolution Governed by Accretive Operators, Dynamical systems vol. 1, Proc. of the Int. Symp. on Dyn. Sys. at Brown U. August 12-16, 1974, L. CESARI, J. K. HALE and J. P. LASALLE, Eds, New York, Academic Press, 1976, pp. 131-165. Zbl0339.35049 MR636953 · Zbl 0339.35049
[13] 13. M. G. CRANDALL, Semigroups of Nonlinear Transformations in Banach Spaces, Contributions to nonlinear Functional Analysis, E. ZARANTONELLO, Ed., New York, Academic Press, 1971, pp. 157-179. Zbl0268.47066 MR470787 · Zbl 0268.47066
[14] 14. M. G. CRANDALL and T. M. LIGGETT, Generation of Semi-groups of Nonlinear Transformations on General Banach Spaces, Amer. J. Math., Vol. 93, 1971, pp. 265-298. Zbl0226.47038 MR287357 · Zbl 0226.47038
[15] 15. A. DAMLAMIAN, Some Results on the Multi-phase Stefan Problem, Comm. on P.D.E., Vol. 2, 1977, pp. 1017-1044. Zbl0399.35054 MR487015 · Zbl 0399.35054
[16] 16. J. DOUGLAS Jr. and T. DUPONT, Alternating-Direction Galerkin Methods on Rectangles, Numerical Solution of Partial Differential Equations-II, SYNSPADE 1970, B. HUBBARD, Ed., New York, Academic press, 1971, pp. 133-214. Zbl0239.65088 MR273830 · Zbl 0239.65088
[17] 17. B. H. GILDING and L. A. PELETIER, On a Class of Similarity Solutions of the Porous Media Equation, J. Math. Anal. Appl., Vol. 55, 1976, pp. 351-364. Zbl0356.35049 MR436751 · Zbl 0356.35049
[18] 18. B. H. GILDING and L. A. PELETIER, On a Class of Similarity Solutions of the Porous Media Equation II, J. Math. Anal. Appl., Vol. 57, 1977, pp. 522-538. Zbl0365.35029 MR436752 · Zbl 0365.35029
[19] 19. D. W. PEACEMAN and H. H. RACHFORD Jr., The Numerical Solution of Parabolic and Elliptic Differential Equations, J. Soc. Indust. Appl. Math., Vol. 3, 1955, pp. 28-41. Zbl0067.35801 MR71874 · Zbl 0067.35801
[20] 20. J. C. W. ROGERS, A. E. BERGER and M. CIMENT, The Alternating Phase Truncation Method for Numerical Solution of a Stefan Problem, to appear in S.I.A.M. J. Num. Anal. Zbl0418.65051 · Zbl 0418.65051
[21] 21. M. ROSE, A Method for Calculating Solutions of Parabolic Equations with a Free Boundary, Math. Comp., Vol. 14, 1960, pp. 249-256. Zbl0096.10102 MR115283 · Zbl 0096.10102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.