An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation

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 results hold for the related Swift–Hohenberg equation as well.