nach oben

BIT Numerical Mathematics

Erschienen in:

10.10.2017

Error estimate of the finite volume scheme for the Allen–Cahn equation

verfasst von: Pavel Strachota, Michal Beneš

Erschienen in: BIT Numerical Mathematics | Ausgabe 2/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The Allen–Cahn equation originates in the phase field formulation of phase transition phenomena. It is a reaction-diffusion ODE with a nonlinear reaction term which allows the formation of a diffuse phase interface. We first introduce a model initial boundary-value problem for the isotropic variant of the equation. Its numerical solution by the method of lines is then considered, using a finite volume scheme for spatial discretization. An error estimate is derived for the solution of the resulting semidiscrete scheme. Subsequently, sample numerical simulations in two and three dimensions are presented and the experimental convergence measurement is discussed.

Vorheriger Artikel Lagrangian and Hamiltonian Taylor variational integrators

Nächster Artikel The effect of uncertain geometries on advection–diffusion of scalar quantities

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)CrossRef

Barrett, J.W., Garcke, H., Nürnberg, R.: On stable parametric finite element methods for the Stefan problem and the Mullins–Sekerka problem with applications to dendritic growth. J. Comput. Phys. 229, 6270–6299 (2010)MathSciNetCrossRefMATH

Beneš, M.: Anisotropic phase-field model with focused latent-heat release. Free Bound. Probl. Theory Appl. II Gakuto Int. Ser. Math. Sci. Appl. 14, 18–30 (2000)MathSciNetMATH

Beneš, M.: Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interface Free Bound. 3, 201–221 (2001)MathSciNetMATH

Beneš, M.: Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math-Czech 48(6), 437–453 (2003)MathSciNetCrossRefMATH

Beneš, M.: Computational studies of anisotropic diffuse interface model of microstructure formation in solidification. Acta Math. Univ. Comen. 76, 39–59 (2007)MathSciNetMATH

Brauer, F., Nohel, J.A.: The Qualitative Theory of Ordinary Differential Equations. Dover Publications, New York (1989)MATH

Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, Chichester (2003)CrossRefMATH

Coudiere, Y., Vila, J.P., Villedieu, P.: Convergence rate of a finite volume scheme for a two-dimensional convection diffusion problem. M2AN Math. Model. Numer. Anal. 33, 493–516 (1999)MathSciNetCrossRefMATH

10.

Drblíková, O., Handlovičová, A., Mikula, K.: Error estimates of the finite volume scheme for the nonlinear tensor-driven anisotropic diffusion. Appl. Numer. Math. 59(10), 2548–2570 (2009). doi:10.1016/j.apnum.2009.05.010 MathSciNetCrossRefMATH

11.

Drblíková, O., Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing. SIAM J. Numer. Anal. 46(1), 37–60 (2007). doi:10.1137/070685038 MathSciNetCrossRefMATH

12.

Elliott, C.M., Gardiner, A.R.: Double obstacle phase field computations of dendritic growth. Tech. Rep. 96/19, University of Sussex at Brighton (1996)

13.

Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. 7, pp. 715–1022. Elsevier, Amsterdam (2000)

14.

Green, J.R., Jimack, P.K., Mullis, A.M., Rosam, J.: An adaptive, multilevel scheme for the implicit solution of three-dimensional phase-field equations. Numer. Meth. Part. D. E. 27, 106–120 (2010)MathSciNetCrossRefMATH

15.

Hartman, P.: Ordinary Differential Equations, Classics in Applied Mathematics, 2nd edn. SIAM, New Delhi (2002)

16.

Hurewicz, W.: Lectures on Ordinary Differential Equations, 2nd edn. M.I.T Press, Cambridge (1970)MATH

17.

Jeong, J.H., Goldenfeld, N., Dantzig, J.A.: Phase field model for three-dimensional dendritic growth with fluid flow. Phys. Rev. E 64, 041602 (2001)CrossRef

18.

Karma, A., Rappel, W.J.: Numerical simulation of three-dimensional dendritic growth. Phys. Rev. Lett. 77(19), 4050–4053 (1996)CrossRef

19.

Karma, A., Rappel, W.J.: Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57(4), 4 (1998)CrossRefMATH

20.

Kupferman, R., Shochet, O., Ben-Jacob, E.: Numerical study of a morphology diagram in the large undercooling limit using a phase-field model. Phys. Rev. E. 50(2), 1005–1008 (1993)CrossRef

21.

Li, M.E., Yang, G.C.: Growth morphologies of a binary alloy with low anisotropy in directional solidification. Acta Metall. Sin. 20, 258–264 (2007)CrossRef

22.

Mullis, A.M., Cochrane, R.F.: A phase field model for spontaneous grain refinement in deeply undercooled metallic melts. Acta Mater. 49, 2205–2214 (2001)CrossRef

23.

Nochetto, R.H., Paolini, M., Verdi, C.: An adaptive finite element method for two-phase Stefan problems in two space dimensions. part I: stability and error estimates. Math. Comput. 57(195), 73–108 (1991)MATH

24.

Nochetto, R.H., Paolini, M., Verdi, C.: An adaptive finite element method for two-phase Stefan problems in two space dimensions. part II: implementation and numerical experiments. J. Sci. Stat. Comput. 12(5), 1207–1244 (1991)CrossRefMATH

25.

Provatas, N., Goldenfeld, N., Dantzig, J.: Adaptive mesh refinement computation of solidification microstructures using dynamic data structures. J. Comput. Phys. 148, 265–290 (1999)MathSciNetCrossRefMATH

26.

PunKay, M.: Modeling of anisotropic surface energies for quantum dot formation and morphological evolution. In: NNIN REU Research Accomplishments, pp. 116–117. University of Michigan (2005)

27.

Rektorys, K.: The Method of Discretization in Time and Partial Differential Equations. D. Reidel Publishing Company, Dordrecht (1982)MATH

28.

Schmidt, A.: Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125, 293–3112 (1996)CrossRefMATH

29.

Singer, H.M., Singer-Loginova, I., Bilgam, J.H., Amberg, G.: Morphology diagram of thermal dendritic solidification by means of phase-field models in two and three dimensions. J. Cryst. Growth. 296, 58–68 (2006)CrossRef

30.

Strachota, P., Beneš, M.: Design and verification of the MPFA scheme for three-dimensional phase field model of dendritic crystal growth. In: A. Cangiani, R.L. Davidchack, E. Georgoulis, A.N. Gorban, J. Levesley, M.V. Tretyakov (eds.) Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, pp. 459–467. Springer Berlin Heidelberg (2013). doi:10.1007/978-3-642-33134-3_49. http://link.springer.com/chapter/10.1007%2F978-3-642-33134-3_49

31.

Strachota, P., Beneš, M., Tintěra, J.: Towards clinical applicability of the diffusion-based DT-MRI visualization algorithm. J. Vis. Commun. Image R. 23(2), 387–396 (2012). doi:10.1016/j.jvcir.2011.11.009 CrossRef

Titel: Error estimate of the finite volume scheme for the Allen–Cahn equation
verfasst von: Pavel Strachota
Michal Beneš
Publikationsdatum: 10.10.2017
Verlag: Springer Netherlands
Erschienen in: BIT Numerical Mathematics / Ausgabe 2/2018
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-017-0687-4

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Weitere Artikel der Ausgabe 2/2018

Evaluation of Chebyshev polynomials by a three-term recurrence in floating-point arithmetic

Composite symmetric general linear methods (COSY-GLMs) for the long-time integration of reversible Hamiltonian systems

General order conditions for stochastic partitioned Runge–Kutta methods

Erratum to: Quasi-interpolation based on the ZP-element for the numerical solution of integral equations on surfaces in

Computing scattering resonances using perfectly matched layers with frequency dependent scaling functions

Optimally zero stable explicit peer methods with variable nodes