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.
Mots clés : phase field crystal, semi-implicit time discretization, convex-concave splitting
@article{M2AN_2013__47_5_1413_0,
author = {Elsey, Matt and Wirth, Benedikt},
title = {A simple and efficient scheme for phase field crystal simulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1413--1432},
publisher = {EDP-Sciences},
volume = {47},
number = {5},
year = {2013},
doi = {10.1051/m2an/2013074},
mrnumber = {3100769},
zbl = {1286.74118},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013074/}
}
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 PY - 2013 SP - 1413 EP - 1432 VL - 47 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013074/ DO - 10.1051/m2an/2013074 LA - en ID - M2AN_2013__47_5_1413_0 ER -
%0 Journal Article %A Elsey, Matt %A Wirth, Benedikt %T A simple and efficient scheme for phase field crystal simulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1413-1432 %V 47 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013074/ %R 10.1051/m2an/2013074 %G en %F M2AN_2013__47_5_1413_0
Elsey, Matt; Wirth, Benedikt. A simple and efficient scheme for phase field crystal simulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1413-1432. doi : 10.1051/m2an/2013074. http://archive.numdam.org/articles/10.1051/m2an/2013074/
[1] , , , and , 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] , and , Inpainting of binary images using the Cahn-Hilliard equation. IEEE Trans. Image Process. 16 (2007) 285-291. | MR | Zbl
[3] and , An efficient algorithm for solving the phase field crystal model. J. Comput. Phys. 227 (2008) 6241-6248. | MR | Zbl
[4] and , Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70 (2004) 051605.
[5] , , and , Modeling elasticity in crystal growth. Phys. Rev. Lett. 88 (2002) 245701.
[6] , , , and , Phase-field crystal modeling and classical density functional theory of freezing. Phys. Rev. B 75 (2007) 064107.
[7] , 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. | MR
[8] , A diffuse interface approach to Hele-Shaw flow. Nonlinearity 16 (2003) 49-66. | MR | Zbl
[9] , , and , Numerical simulation of grain-boundary grooving by level set method. J. Comput. Phys. 170 (2001) 764-784. | Zbl
[10] and , On the remainder term in the N-dimensional Euler Maclaurin expansion. Numer. Math. 15 (1970) 333-344. | MR | Zbl
[11] , 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. | MR | Zbl
[12] , and , An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47 (2009) 2269-2288. | MR | Zbl
[13] and , Phase-field crystal modeling of equilibrium bcc-liquid interfaces. Phys. Rev. B 76 (2007) 184107.
[14] , and , Controlling crystal symmetries in phase-field crystal models. J. Phys. Condensed Matter 22 (2010) 364102.
[15] and , Stress-induced morphological instabilities at the nanoscale examined using the phase field crystal approach. Phys. Rev. B 80 (2009) 125408.
[16] , , and , Density-amplitude formulation of the phase-field crystal model for two-phase coexistence in two and three dimensions. Philosophical Magazine 90 (2010) 237-263.
Cité par Sources :
