Journal Article

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

WANG, C., WISE, S. M.

2011

Overview

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.

• Publisher: Society for Industrial and Applied Mathematics
• Subject: 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