On the numerical approach of the enthalpy method for the Stefan problem

Numerical Methods for Partial Differential Equations

Volumn 20, Issue 4, 2004, Pages 527-551

  Omrani, Khaled ^a  

^a Inst Sup Sci Appl T Sousse   (Tunisia)

Author keywords

Convergence; Error estimate; Finite difference; Finite element; Stefan problem

Indexed keywords

EID: 2942737438     PISSN: 0749159X     EISSN: None     Source Type: Journal    
DOI: 10.1002/num.10105     Document Type: Article

References (35)

1. 1 V. Alexiades and A. D. Solomon, Mathematical modeling of melting and freezing processes, Hemisphere, Taylor & Francis, Washington, DC, 1993.
2. 2 A. A. Samarskii, P. N. Vabishchevich, O. P. Iliev, and A. G. Churbanov, Numerical simulation of convection‐diffusion phase change problems, Int J Heat Mass Transfer 36 (17) (1993), 4095–4106.
3. 3 A. E. Berger, H. Brezis, and J. C. W. Rogers, A numerical method for solving the problem u t − Δf(u), RAIRO Anal Numer 13 (1979), 297–312.
4. 4 H. Brezis, On some degenerate nonlinear parabolic equations, nonlinear functional analysis, F. Browder, editor, Symp in Pure Mathematics, Vol. 18, American Meteorological Society, Providence, 1970, pp. 28–38.
5. 5 J. Douglas, Jr. and T. Dupont, Galerkin methods for parabolic equations, SIAM J Numer Anal 7 (1970), 575–626.
6. 6 J. W. Jerome, Approximation of nonlinear evolution systems, Academic Press, New York, 1983.
7. 7 S. Kamenomostskaja, (in Russian) On the Stefan problem, Mat Sb 53 (1961), 489–514.
8. 8 R. H. Nochetto, Error estimates for two‐phase Stefan problems in sevral space variables, I: linear boundary conditions, Calcolo 22 (1985), 457–499.
9. 9 R. H. Nochetto, Error estimates for two‐phase Stefan problems in sevral space variables, II: nonlinear flux conditions, Calcolo 22 (1985), 501–534.
10. 10 R. H. Nochetto, Error estimates for multidimensional singular parabolic problems, Jpn J Appl Math 4 (1987), 111–138.
11. 11 C. M. Elliott, Error analysis of the enthalpy method for the Stefan problem, IMA J Numer Anal 7 (1987), 61–71.
12. 12 J. W. Jerome and M. E. Rose, Error estimates for the multidimensional two‐phase Stefan problem, Math Comp 39 (1982), 377–414.
13. 13 C. Verdi, On the numerical approach to a two‐phase Stefan problem with nonlinear flux, Calcolo 22 (1985), 351–381.
14. 14 V. R. Voller and M. Cross, Application of control volume enthalpy methods in the solution of Stefan problems, R. W. Lewis, editor, Recent advances in computational heat transfer, Pineridge Press, 1984.
15. 15 R. E. White, A numerical solution of the enthalpy formulation of the Stefan problem, SIAM J Numer Anal 19 (1982), 1129–1157.
16. 16 J. F. Epperson, Finite element methods for a class of nonlinear evolution equations, SIAM J Numer Anal 21 (1984), 1066–1079.
17. 17 J. F. Epperson, Regularization for aclass of nonlinear evolution equations, SIAM J Math Anal 17 (1986), 84–90.
18. 18 R. H. Nochetto, Error estimates for multidimensional Stefan problems with general boundary conditions, Proceedings of the Conference on Free Boundary Problems, June 1984, Maubuisson, France. A. Bossavit, A. Damlamian and M. Frémond, editors, Pitman, London, 1985.
19. 19 C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, Pitman, London, 1982.
20. 20 J. Crank, Free and moving boundary problems, Clarendon Press, Oxford, 1984.
21. 21 J. Crank, A nostalgic look at the mechanical solution of Stefan problems, Bull IMA 11 (1975), 22–33.
22. 22 J. L. Lions, Quelques méthodes de resolutions des problèmes aux limites non linéaires, Gauthier Villars Paris, 1969.
23. 23 J. F. Ciavaldini, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'élements finis, SIAM J Numer Anal 12 (1975), 464–487.
24. 24 V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics 1054, Springer, New York, 1984.
25. 25 M. F. Wheeler, A priori L 2 ‐error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J Numer Anal 10 (1973), 723–759.
26. 26 J. Douglas, Jr., T. Dupont, R. E. Ewing, Incomplete iteration for time‐stepping a Galerkin method for a quasilinear parabolic problem, SIAM J Numer Anal 16 (1979), 503–522.
27. 27 R. E. Ewing, Efficient multistep procedures for nonlinear parabolic problems with nonlinear Neumann boundary conditions, Calcolo 19 (1982), 231–252.
28. 28 M. Luskin, A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions, SIAM J Numer Anal 16 (1979), 284–299.
29. 29 H. B. Keller, A new difference scheme for parabolic problems, B. Hubbard, editor, Numerical solution of partial differential equations II, Academic Press, New York, 1971.
30. 30 S. Zhi‐zhong, A second‐order accurate linearized difference scheme for the two‐dimensional Cahn‐Hilliard equation, Math Comput 64 (1995), N 212.
31. 31 F. E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, R. Finn, editor, Applications of nonlinear P.D.Es, Proc Symp Appl Math, Vol. 17, American Meteorological Society, Providence, 1965, pp. 24–49.
32. 32 K. Omrani, On L ∞ convergence of difference scheme for the Cahn‐Hilliard equation, J Comput Math (2003), submitted.
33. 33 A. Damlamian, Some results on the multi‐phase Stefan problem, Common PDE 2 (1977), 1017–1044.
34. 34 K. Omrani, A second order splitting method for a finite difference scheme for the Sivashinsky equation, Appl Math Lett 16 (2003), 441–445.
35. 35 V. R. Voller, Implicit finite‐difference solutions of the enthalpy formulation of Stefan problem, IMA J Numer Anal 5 (1985), 201–214.