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In this paper we propose and analyze an energy stable numerical scheme forthe square phase field crystal (SPFC) equation, a gradient flow modelingcrystal dynamics at the atomic scale in space but on diffusive scales in time.In particular, a modification of the free energy potential to the standardphase field crystal model leads to a composition of the 4-Laplacian and theregular Laplacian operators. To overcome the difficulties associated with thishighly nonlinear operator, we design numerical algorithms based on thestructures of the individual energy terms. A Fourier pseudo-spectralapproximation is taken in space, in such a way that the energy structure isrespected, and summation-by-parts formulae enable us to study the discreteenergy stability for such a high-order spatial discretization. In the temporalapproximation, a second order BDF stencil is applied, combined with anappropriate extrapolation for the concave diffusion term(s). A second orderartificial Douglas-Dupont-type regularization term is added to ensure energystability, and a careful analysis leads to the artificial linear diffusioncoming at an order lower that that of surface diffusion term. Such a choiceleads to reduced numerical dissipation. At a theoretical level, the uniquesolvability, energy stability are established, and an optimal rate convergenceanalysis is derived in the $\ell^\infty (0,T; \ell^2) \cap \ell^2 (0,T; H_N^3)$norm. In the numerical implementation, the preconditioned steepest descent(PSD) iteration is applied to solve for the composition of the highly nonlinear4-Laplacian term and the standard Laplacian term, and a geometric convergenceis assured for such an iteration. Finally, a few numerical experiments arepresented, which confirm the robustness and accuracy of the proposed scheme.

An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation