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

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 a...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2011-01, Vol.49 (3/4), p.945-969
Main Authors: WANG, C., WISE, S. M.
Format: Article
Language:English
Subjects:
Analytical estimating
Cauchy Schwarz inequality
Conservation of mass
Convergence
Crystals
Differential equations
Direct power generation
Estimates
Estimation methods
Exact sciences and technology
Finite difference method
Inner products
Mathematical analysis
Mathematical models
Mathematics
Nonlinear equations
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Periodic functions
Sciences and techniques of general use
Splitting
Studies
Thin films
Truncation errors
Wave equations
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Summary: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.
ISSN:0036-1429
1095-7170
DOI:10.1137/090752675