Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T16:08:19.190Z Has data issue: false hasContentIssue false
  • English
  • Français

An implicit scheme to solve a system of ODEs arising from the space discretization of nonlinear diffusion equations

Published online by Cambridge University Press:  15 April 2002

Éric Boillat
Éric Boillat*
Affiliation:
Department of Mathematics, EPFL, 1015 Lausanne, Switzerland. (eric.boillat@epfl.ch)
Get access

Abstract

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.

Keywords

Nonlinear diffusion equationsnonlinear parabolic problemChernoff schemeimplicit scheme for ODE's.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 4 , July 2001 , pp. 749 - 765
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Friedman, A., The Stefan problem in several space variables. Trans. Amer. Math. Soc. 132 (1968) 51-87. CrossRef
Berger, A.E., Brezis, H. and Rogers, J.C.W., A numerical method for solving ut - Δƒ(u) = 0. RAIRO. Anal. Numér. 13 (1979) 297-312. CrossRef
Elliott, C.M., Error analysis of the enthalpy method for the Stefan problem. IMA J. Numer. Anal. 7 (1987) 61-71. CrossRef
S.R. De Groot and P. Mazur, Non-equilibrium thermodynamics. North-Holland, Amsterdam (1962).
H. Brezis, Analyse fonctionnelle, Théorie et applications. Masson, Paris (1993).
Alt, H.W. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983) 311-341.
I. Prigogine, Thermodynamics of irreversible processes. Interscience Publ. (1967).
Donnelly, J.D.P., A model for non-equilibrium thermodynamic processes involving phase changes. J. Inst. Math. Appl. 24 (1979) 425-438. CrossRef
Ciavaldini, J.F., Analyse numérique d'un problème de Stefan à deux phases par une méthode d'éléments finis. SIAM J. Numer. Anal. 12 (1975) 464-487. CrossRef
Jerome, J.W. and Rose, M.E., Error estimates for the multidimensional two-phase Stefan problem. Math. Comp. 39 (1982) 377-414. CrossRef
K. Yosida, Functional Analysis. Springer-Verlag, Berlin (1984).
E. Magenes, Remarques sur l'approximation des problèmes paraboliques non-linéaires, in Analyse Mathématique et Applications, Gauthier-Villars, Paris (1988) 297-318.
Magenes, E., Nochetto, R.H. and Verdi, C., Energy error estimates for a linear scheme to approximate nonlinear parabolic problems. RAIRO. Modèl. Math. Anal. Numér. 21 (1987) 655-678. CrossRef
M. Crouzeix and A.L. Mignot, Analyse numérique des équations différentielles. Masson (1989).
Meyer, G.H., Multidimensional Stefan problems. SIAM J. Numer. Anal. 10 (1973) 522-538. CrossRef
O. Krüger, Modélisation et analyse numérique de problèmes de réaction-diffusion provenant de la solidification d'alliages binaires. Technical Report 2071, Thèse EPFL (1999).
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985).
Paolini, M., Sacchi, G. and Verdi, C., Finite element approximations of singular parabolic problems. Internat. J. Numer. Methods Engrg. 26 (1988) 1989-2007. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problem. North Holland, Amsterdam (1978).
Rulla, J., Error analysis for implicit approximations to Cauchy problems. SIAM J. Numer. Anal. 33 (1996) 68-87. CrossRef
V. Thomée, Galerkin finite element methods for Parabolic Problems. Springer-Verlag, Berlin (1984).
Jäger, W. and Kačur, J., Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes. RAIRO. Modèl. Math. Anal. Numér. 29 (1995) 605-627. CrossRef