A simple and efficient scheme for phase field crystal simulation

Matt Elsey; Benedikt Wirth

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1413-1432
  • ISSN: 0764-583X

Abstract

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We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error analysis as well as numerical experiments to validate the method’s efficiency.

How to cite

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Elsey, Matt, and Wirth, Benedikt. "A simple and efficient scheme for phase field crystal simulation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1413-1432. <http://eudml.org/doc/273322>.

@article{Elsey2013,
abstract = {We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error analysis as well as numerical experiments to validate the method’s efficiency.},
author = {Elsey, Matt, Wirth, Benedikt},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {phase field crystal; semi-implicit time discretization; convex-concave splitting; embedding; convergence},
language = {eng},
number = {5},
pages = {1413-1432},
publisher = {EDP-Sciences},
title = {A simple and efficient scheme for phase field crystal simulation},
url = {http://eudml.org/doc/273322},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Elsey, Matt
AU - Wirth, Benedikt
TI - A simple and efficient scheme for phase field crystal simulation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1413
EP - 1432
AB - We propose an unconditionally stable semi-implicit time discretization of the phase field crystal evolution. It is based on splitting the underlying energy into convex and concave parts and then performing H-1 gradient descent steps implicitly for the former and explicitly for the latter. The splitting is effected in such a way that the resulting equations are linear in each time step and allow an extremely simple implementation and efficient solution. We provide the associated stability and error analysis as well as numerical experiments to validate the method’s efficiency.
LA - eng
KW - phase field crystal; semi-implicit time discretization; convex-concave splitting; embedding; convergence
UR - http://eudml.org/doc/273322
ER -

References

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  1. [1] B.P. Athreya, N. Goldenfeld, J.A. Dantzig, M. Greenwood and N. Provatas, Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model. Phys. Rev. E 76 (2007) 056706. 
  2. [2] A.L. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans. Image Process.16 (2007) 285–291. Zbl1279.94008MR2460167
  3. [3] M. Cheng and J.A. Warren, An efficient algorithm for solving the phase field crystal model. J. Comput. Phys.227 (2008) 6241–6248. Zbl1151.82411MR2418360
  4. [4] K.R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70 (2004) 051605. 
  5. [5] K.R. Elder, M. Katakowski, M. Haataja and M. Grant, Modeling elasticity in crystal growth. Phys. Rev. Lett. 88 (2002) 245701. 
  6. [6] K.R. Elder, N. Provatas, J. Berry, P. Stefanovic and M. Grant, Phase-field crystal modeling and classical density functional theory of freezing. Phys. Rev. B 75 (2007) 064107. 
  7. [7] D. Eyre, Unconditionally gradient stable time marching the Cahn–Hilliard equation, in Computational and mathematical models of microstructural evolution, edited by J.W. Bullard, R. Kalia, M. Stoneham and L.Q. Chen. Warrendale, PA, Materials Research Society 53 (1998) 1686–1712. Zbl0853.73060MR1676409
  8. [8] K. Glasner, A diffuse interface approach to Hele–Shaw flow. Nonlinearity16 (2003) 49–66. Zbl1138.76340MR1950775
  9. [9] M. Khenner, A. Averbuch, M. Israeli and M. Nathan, Numerical simulation of grain-boundary grooving by level set method. J. Comput. Phys.170 (2001) 764–784. Zbl1112.74457
  10. [10] J.N. Lyness and B.J.J. McHugh, On the remainder term in the N-dimensional Euler Maclaurin expansion. Numer. Math.15 (1970) 333–344. Zbl0199.11801MR267734
  11. [11] P. Smereka, Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19 (2003) 439–456. Special issue in honor of the sixtieth birthday of Stanley Osher. Zbl1035.65098MR2028853
  12. [12] S.M. Wise, C. Wang and J.S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal.47 (2009) 2269–2288. Zbl1201.35027MR2519603
  13. [13] K.-A. Wu and A. Karma, Phase-field crystal modeling of equilibrium bcc-liquid interfaces. Phys. Rev. B 76 (2007) 184107. 
  14. [14] K.-A. Wu, M. Plapp and P.W. Voorhees, Controlling crystal symmetries in phase-field crystal models. J. Phys. Condensed Matter 22 (2010) 364102. 
  15. [15] K.-A. Wu and P.W. Voorhees, Stress-induced morphological instabilities at the nanoscale examined using the phase field crystal approach. Phys. Rev. B 80 (2009) 125408. 
  16. [16] D.-H. Yeon, Z.-F. Huang, K. Elder and K. Thornton, Density-amplitude formulation of the phase-field crystal model for two–phase coexistence in two and three dimensions. Philosophical Magazine90 (2010) 237–263. 

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