AN ENERGY STABLE AND CONVERGENT FINITE-DIFFERENCE SCHEME FOR THE MODIFIED PHASE FIELD CRYSTAL EQUATION

被引：276

作者：

Wang, C. ^{[1
]}

Wise, S. M. ^{[2
]}

机构：

[1] Univ Massachusetts, Dept Math, N Dartmouth, MA 02747 USA

[2] Univ Tennessee, Dept Math, Knoxville, TN 37996 USA

来源：

SIAM JOURNAL ON NUMERICAL ANALYSIS | 2011年 / 49卷 / 03期

基金：

美国国家科学基金会;

关键词：

Phase Field Crystal; Modified Phase Field Crystal; finite-difference methods; stability; nonlinear partial differential equations;

D O I：

10.1137/090752675

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We present an unconditionally energy stable finite difference scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The method is based on a convex splitting of a discrete pseudoenergy and is semi-implicit. The equation at the implicit time level is nonlinear but represents the gradient of a strictly convex function and is thus uniquely solvable, regardless of time step-size. We present a local-in-time error estimate that ensures the pointwise convergence of the scheme.

引用

页码：945 / 969

页数：25

共 16 条

[1] Nucleation and growth by a phase field crystal (PFC) model [J].

Backofen, R. ;

Raetz, A. ;

Voigt, A. .

PHILOSOPHICAL MAGAZINE LETTERS, 2007, 87 (11) :813-820

[2] A 2ND ORDER FINITE DIFFERENCE ANALOG OF FIRST BIHARMONIC BOUNDARY VALUE PROBLEM [J].

BRAMBLE, JH .

NUMERISCHE MATHEMATIK, 1966, 9 (03) :236-&

[3]

CHEN W, J SCI COMPUT UNPUB

[4] Modeling elasticity in crystal growth [J].

Elder, KR ;

Katakowski, M ;

Haataja, M ;

Grant, M .

PHYSICAL REVIEW LETTERS, 2002, 88 (24) :2457011-2457014

[5] Unconditionally gradient stable time marching the Cahn-Hilliard equation [J].

Eyre, DJ .

COMPUTATIONAL AND MATHEMATICAL MODELS OF MICROSTRUCTURAL EVOLUTION, 1998, 529 :39-46

[6]

Furihata D, 2001, NUMER MATH, V87, P675, DOI 10.1007/s002110000212

[7] Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation [J].

Hu, Z. ;

Wise, S. M. ;

Wang, C. ;

Lowengrub, J. S. .

JOURNAL OF COMPUTATIONAL PHYSICS, 2009, 228 (15) :5323-5339

[8] Dynamic density functional theory of fluids [J].

Marconi, UMB ;

Tarazona, P .

JOURNAL OF CHEMICAL PHYSICS, 1999, 110 (16) :8032-8044

[9] Using the phase-field crystal method in the multi-scale modeling of microstructure evolution [J].

Provatas, N. ;

Dantzig, J. A. ;

Athreya, B. ;

Chan, P. ;

Stefanovic, P. ;

Goldenfeld, N. ;

Elder, K. R. .

JOM, 2007, 59 (07) :83-90

[10]

SHEN J, SIAM J NUMER A UNPUB

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