An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation (Communications in Computational Physics, 2019)
In this paper we propose and analyze an energy stable numerical schemefor the square phase field crystal (SPFC) equation, a gradient flow modeling crystaldynamics at the atomic scale in space but on diffusive scales in time. In particular, amodification of the free energy potential to the standard phase field crystal model leadsto a composition of the 4-Laplacian and the regular Laplacian operators. To overcomethe difficulties associated with this highly nonlinear operator, we design numericalalgorithms based on the structures of the individual energy terms. A Fourier pseudo-spectral approximation is taken in space, in such a way that the energy structure isrespected, and summation-by-parts formulae enable us to study the discrete energystability for such a high-order spatial discretization. In the temporal approximation, asecond order BDF stencil is applied, combined with an appropriate extrapolation forthe concave diffusion term(s). A second order artificial Douglas-Dupont-type regularization term is added to ensure energy stability, and a careful analysis leads to theartificial linear diffusion coming at an order lower than that of surface diffusion term.Such a choice leads to reduced numerical dissipation. At a theoretical level, the uniquesolvability, energy stability are established, and an optimal rate convergence analysisis derived in the $ℓ$∞(0,$T$;$ℓ$2)∩$ℓ$2(0,$T$;$H_N^3$) norm. In the numerical implementation, thepreconditioned steepest descent (PSD) iteration is applied to solve for the composition of the highly nonlinear 4-Laplacian term and the standard Laplacian term, and ageometric convergence is assured for such an iteration. Finally, a few numerical experiments are presented, which confirm the robustness and accuracy of the proposedscheme.