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Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation

Authors:

J. S. LowengrubAuthors Info & Claims

Journal of Computational Physics, Volume 228, Issue 15

Pages 5323 - 5339

https://doi.org/10.1016/j.jcp.2009.04.020

Published: 01 August 2009 Publication History

Abstract

In this paper we present and compare two unconditionally energy stable finite-difference schemes for the phase field crystal equation. The first is a one-step scheme based on a convex splitting of a discrete energy by Wise et al. [S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press]. In this scheme, which is first order in time and second order in space, the discrete energy is non-increasing for any time step. The second scheme we consider is a new, fully second-order two-step algorithm. In the new scheme, the discrete energy is bounded by its initial value for any time step. In both methods, the equations at the implicit time level are nonlinear but represent the gradients of strictly convex functions and are thus uniquely solvable, regardless of time step-size. We solve the nonlinear equations using an efficient nonlinear multigrid method. Numerical simulations are presented and confirm the stability, efficiency and accuracy of the schemes.

References

[1]

Backofen, R., Rätz, A. and Voigt, A., Nucleation and growth by a phase field crystal (PFC) model. Phil. Mag. Lett. v87. 813

[2]

Cheng, M. and Warren, J.A., An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. v227. 6241

[3]

Elder, K.R. and Grant, M., Modeling elastic and plastic deformations in nonequilibrium processing using phase filed crystal. Phys. Rev. E. v90. 051605

[4]

Elder, K.R., Katakowski, M., Haataja, M. and Grant, M., Modeling elasticity in crystal growth. Phys. Rev. Lett. v88. 245701

[5]

D. Eyre, Unconditionally gradient stable time marching the Cahn-Hilliard equation, in: J.W. Bullard, R. Kalia, M. Stoneham, L.Q. Chen (Eds.), Computational and Mathematical Models of Microstructural Evolution, vol. 53, Materials Research Society, Warrendale, PA, USA, 1998, pp. 1686-1712.

[6]

He, Y., Liu, Y. and Tang, T., On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. v57 i4. 616-628.

[7]

Z. Hu, S. Wise, C. Wang, J.S. Lowengrub, Stable and efficient finite-difference nonlinear-multigrid schemes for the modified phase field crystal equation, in preparation.

[8]

Kim, J.S., Kang, K. and Lowengrub, J.S., Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. v193. 511-543.

Digital Library

[9]

Leo, P.H., Lowengrub, J.S. and Jou, H.J., A diffuse interface model for microstructural evolution in elastically stressed solids. Acta Mater. v46. 2113

[10]

Marconi, U.M.B. and Tarazona, P., Dynamic density functional theory of liquids. J. Chem. Phys. v110. 8032

[11]

Mellenthin, J., Karma, A. and Plapp, M., Phase-field crystal study of grain-boundary premelting. Phys. Rev. B. v78. 184110

[12]

Haataja, M., Stefanovic, P. and Provatas, N., Phase-field crystals with elastic interactions. Phys. Rev. Lett. v96. 225504

[13]

Provatas, N., Dantzig, J.A., Athreya, B., Chan, P., Stefanovic, P., Goldenfeld, N. and Elder, K.R., Using the phase-field crystal method in the multiscale modeling of microstructure evolution. JOM. v59. 83

[14]

Swift, J. and Hohenberg, P.C., Hydrodynamic fluctuations at the convective instability. Phys. Rev. A. v15. 319

[15]

Trottenberg, U., Oosterlee, C.W. and Schüller, A., Multigrid. 2005. Academic Press, New York.

[16]

Vollmayr-Lee, B.P. and Rutenberg, A.D., Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E. v68. 066703

[17]

C. Wang, S. Wise, An energy stable and convergent finite-difference scheme for the modified phase field crystal equation, SIAM J. Numer. Anal., submitted for publication.

[18]

Wise, S.M., Kim, J.S., Lowengrub, J.S. and regularized, Solving the, strongly anisotropic Chan-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. v226. 414-446.

[19]

S.M. Wise, C. Wang, J.S. Lowengrub, An energy stable and convergent finite-difference scheme for the phase field crystal equation, SIAM J. Numer. Anal., in press.

[20]

Xu, C. and Tang, T., Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. v44. 1759-1779.

Cited By

Qi L(2024)Numerical analysis of a second-order energy-stable finite element method for the Swift-Hohenberg equationApplied Numerical Mathematics10.1016/j.apnum.2023.11.014197:C(119-142)Online publication date: 1-Mar-2024
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Guo YYin QZhang Z(2024)A Structure-preserving Implicit Exponential Time Differencing Scheme for Maxwell–Ampère Nernst–Planck ModelJournal of Scientific Computing10.1007/s10915-024-02669-0101:2Online publication date: 10-Oct-2024
https://dl.acm.org/doi/10.1007/s10915-024-02669-0
Xie YLi QMei L(2024)An Unconditionally Energy Stable Method for the Anisotropic Phase-Field Crystal Model in Two DimensionJournal of Scientific Computing10.1007/s10915-024-02543-z100:1Online publication date: 28-May-2024
https://dl.acm.org/doi/10.1007/s10915-024-02543-z
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Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations

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Published In

cover image Journal of Computational Physics

Journal of Computational Physics Volume 228, Issue 15

August, 2009

306 pages

ISSN:0021-9991

Issue’s Table of Contents

Copyright © Elsevier Inc. © 2009.

Publisher

Academic Press Professional, Inc.

United States

Publication History

Published: 01 August 2009

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Cited By

Qi L(2024)Numerical analysis of a second-order energy-stable finite element method for the Swift-Hohenberg equationApplied Numerical Mathematics10.1016/j.apnum.2023.11.014197:C(119-142)Online publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1016/j.apnum.2023.11.014
Guo YYin QZhang Z(2024)A Structure-preserving Implicit Exponential Time Differencing Scheme for Maxwell–Ampère Nernst–Planck ModelJournal of Scientific Computing10.1007/s10915-024-02669-0101:2Online publication date: 10-Oct-2024
https://dl.acm.org/doi/10.1007/s10915-024-02669-0
Xie YLi QMei L(2024)An Unconditionally Energy Stable Method for the Anisotropic Phase-Field Crystal Model in Two DimensionJournal of Scientific Computing10.1007/s10915-024-02543-z100:1Online publication date: 28-May-2024
https://dl.acm.org/doi/10.1007/s10915-024-02543-z
Zhang DZhang L(2023)Spectral deferred correction method for Landau–Brazovskii model with convex splitting techniqueJournal of Computational Physics10.1016/j.jcp.2023.112348491:COnline publication date: 15-Oct-2023
https://dl.acm.org/doi/10.1016/j.jcp.2023.112348
Hou DNing YZhang C(2023)An efficient and robust Lagrange multiplier approach with a penalty term for phase-field modelsJournal of Computational Physics10.1016/j.jcp.2023.112236488:COnline publication date: 1-Sep-2023
https://dl.acm.org/doi/10.1016/j.jcp.2023.112236
Liu ZChen C(2023)On efficient semi-implicit auxiliary variable methods for the six-order Swift–Hohenberg modelJournal of Computational and Applied Mathematics10.1016/j.cam.2022.114730419:COnline publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1016/j.cam.2022.114730
Qi LHou Y(2023)Error estimates for the Scalar Auxiliary Variable (SAV) schemes to the modified phase field crystal equationJournal of Computational and Applied Mathematics10.1016/j.cam.2022.114579417:COnline publication date: 1-Jan-2023
https://dl.acm.org/doi/10.1016/j.cam.2022.114579
Liu CWang CWise SYue XZhou S(2023)A Second Order Accurate, Positivity Preserving Numerical Method for the Poisson–Nernst–Planck System and Its Convergence AnalysisJournal of Scientific Computing10.1007/s10915-023-02345-997:1Online publication date: 12-Sep-2023
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Zou GLi ZYang X(2023)Fully Discrete Discontinuous Galerkin Numerical Scheme with Second-Order Temporal Accuracy for the Hydrodynamically Coupled Lipid Vesicle ModelJournal of Scientific Computing10.1007/s10915-023-02129-195:1Online publication date: 15-Feb-2023
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Pei SHou YLi Q(2022)A linearly second-order, unconditionally energy stable scheme and its error estimates for the modified phase field crystal equationComputers & Mathematics with Applications10.1016/j.camwa.2021.10.029103:C(104-126)Online publication date: 23-Apr-2022
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