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In this paper we propose and analyze a second order accurate (in time)numerical scheme for the square phase field crystal (SPFC) equation, a gradientflow modeling crystal dynamics at the atomic scale in space but on diffusivescales in time. Its primary difference with the standard phase field crystalmodel is an introduction of the 4-Laplacian term in the free energy potential,which in turn leads to a much higher degree of nonlinearity. To make thenumerical scheme linear while preserving the nonlinear energy stability, wemake use of the scalar auxiliary variable (SAV) approach, in which a secondorder backward differentiation formula (BDF) is applied in the temporalstencil. Meanwhile, a direct application of the SAV method faces certaindifficulties, due to the involvement of the 4-Laplacian term, combined with aderivation of the lower bound of the nonlinear energy functional. In theproposed numerical method, an appropriate decomposition for the physical energyfunctional is formulated, so that the nonlinear energy part has awell-established global lower bound, and the rest terms lead toconstant-coefficient diffusion terms with positive eigenvalues. In turn, thenumerical scheme could be very efficiently implemented by constant-coefficientPoisson-like type solvers (via FFT), and energy stability is established byintroducing an auxiliary variable, and an optimal rate convergence analysis isprovided for the proposed SAV method. A few numerical experiments are alsopresented, which confirm the efficiency and accuracy of the proposed scheme.

A second order accurate scalar auxiliary variable (SAV) numerical method for the square phase field crystal equation