  In this paper we propose and analyze a second order accurate (in time)
numerical scheme for the square phase field crystal (SPFC) equation, a gradient
flow modeling crystal dynamics at the atomic scale in space but on diffusive
scales in time. Its primary difference with the standard phase field crystal
model 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 the
numerical scheme linear while preserving the nonlinear energy stability, we
make use of the scalar auxiliary variable (SAV) approach, in which a second
order backward differentiation formula (BDF) is applied in the temporal
stencil. Meanwhile, a direct application of the SAV method faces certain
difficulties, due to the involvement of the 4-Laplacian term, combined with a
derivation of the lower bound of the nonlinear energy functional. In the
proposed numerical method, an appropriate decomposition for the physical energy
functional is formulated, so that the nonlinear energy part has a
well-established global lower bound, and the rest terms lead to
constant-coefficient diffusion terms with positive eigenvalues. In turn, the
numerical scheme could be very efficiently implemented by constant-coefficient
Poisson-like type solvers (via FFT), and energy stability is established by
introducing an auxiliary variable, and an optimal rate convergence analysis is
provided for the proposed SAV method. A few numerical experiments are also
presented, which confirm the efficiency and accuracy of the proposed scheme.
