no. 4
An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 749-765.

In this article, we consider the initial value problem which is obtained after a space discretization (with space step h) of the equations governing the solidification process of a multicomponent alloy. We propose a numerical scheme to solve numerically this initial value problem. We prove an error estimate which is not affected by the step size h chosen in the space discretization. Consequently, our scheme provides global convergence without any stability condition between h and the time step size τ. Moreover, it is not of excessive algorithmic complexity since it does not require more than one resolution of a linear system at each time step.

Classification : 65L05, 65L80, 65N30
Mots clés : nonlinear diffusion equations, nonlinear parabolic problem, Chernoff scheme, implicit scheme for ODE's 
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     author = {Boillat, \'Eric},
     title = {An implicit scheme to solve a system of {ODEs} arising from the space discretization of nonlinear diffusion equations},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {749--765},
     publisher = {EDP-Sciences},
     volume = {35},
     number = {4},
     year = {2001},
     mrnumber = {1863278},
     zbl = {0991.65091},
     language = {en},
     url = {http://archive.numdam.org/item/M2AN_2001__35_4_749_0/}
}
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Boillat, Éric. An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 4, pp. 749-765. http://archive.numdam.org/item/M2AN_2001__35_4_749_0/

[1] A. Friedman, The Stefan problem in several space variables. Trans. Amer. Math. Soc. 132 (1968) 51-87. | MR | Zbl

[2] A.E. Berger, H. Brezis and J.C.W. Rogers, A numerical method for solving u t -Δf(u)=0. RAIRO. Anal. Numér. 13 (1979) 297-312. | Numdam | MR | Zbl

[3] C.M. Elliott, Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7 (1987) 61-71. | MR | Zbl

[4] S.R. De Groot and P. Mazur, Non-equilibrium thermodynamics. North-Holland, Amsterdam (1962).

[5] H. Brezis, Analyse fonctionnelle, Théorie et applications. Masson, Paris (1993). | MR | Zbl

[6] H.W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341. | MR | Zbl

[7] I. Prigogine, Thermodynamics of irreversible processes. Interscience Publ. (1967).

[8] J.D.P. Donnelly, A model for non-equilibrium thermodynamic processes involving phase changes. J. Inst. Math. Appl. 24 (1979) 425-438. | MR | Zbl

[9] J.F. Ciavaldini, Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464-487. | Zbl

[10] J.W. Jerome and M.E. Rose, Error estimates for the multidimensional two-phase Stefan problem. Math. Comp. 39 (1982) 377-414. | MR | Zbl

[11] K. Yosida, Functional Analysis. Springer-Verlag, Berlin (1984).

[12] E. Magenes, Remarques sur l'approximation des problèmes paraboliques non-linéaires, in Analyse Mathématique et Applications, Gauthier-Villars, Paris (1988) 297-318. | Zbl

[13] E. Magenes, R.H. Nochetto and C. Verdi, Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. RAIRO. Modèl. Math. Anal. Numér. 21 (1987) 655-678. | Numdam | MR | Zbl

[14] M. Crouzeix and A.L. Mignot, Analyse numérique des équations différentielles. Masson (1989). | MR | Zbl

[15] G.H. Meyer, Multidimensional Stefan problems. SIAM J. Numer. Anal. 10 (1973) 522-538. | MR | Zbl

[16] O. Krüger, Modélisation et analyse numérique de problèmes de réaction-diffusion provenant de la solidification d'alliages binaires. Technical Report 2071, Thèse EPFL (1999).

[17] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR | Zbl

[18] M. Paolini, G. Sacchi and C. Verdi, Finite element approximations of singular parabolic problems. Internat. J. Numer. Methods Engrg. 26 (1988) 1989-2007. | MR | Zbl

[19] P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978). | MR | Zbl

[20] J. Rulla, Error analysis for implicit approximations to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 68-87. | MR | Zbl

[21] V. Thomée, Galerkin finite element methods for Parabolic Problems. Springer-Verlag, Berlin (1984). | MR | Zbl

[22] W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO. Modèl. Math. Anal. Numér. 29 (1995) 605-627. | Numdam | MR | Zbl