We present an unconditionally energy stable finite-difference scheme for the phase field
crystal equation. The method is based on a convex splitting of a discrete energy 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 local-in-time error estimates that ensure the convergence of the scheme. While this
paper is primarily concerned with the phase field crystal equation, most of the theoretical …

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 …