A Linear Adaptive Second-order Backward Differentiation Formulation Scheme for the Phase Field Crystal Equation

In this article, we present and analyze a linear fully discrete second order scheme with variable time steps for the phase field crystal equation. More precisely, we construct a linear adaptive time stepping scheme based on the second order backward differentiation formulation (BDF2) and use the Fourier spectral method for the spatial discretization. The scalar auxiliary variable approach is employed to deal with the nonlinear term, in which we only adopt a first order method to approximate the auxiliary variable. This treatment is extremely important in the derivation of the unconditional energy stability of the proposed adaptive BDF2 scheme. However, we find for the first time that this strategy will not affect the second order accuracy of the unknown phase function (Formula presented.) by setting the positive constant (Formula presented.) large enough such that (Formula presented.) The energy stability of the adaptive BDF2 scheme is established with a mild constraint on the adjacent time step radio (Formula presented.). Furthermore, a rigorous error estimate of the second order accuracy of (Formula presented.) is derived for the proposed scheme on the nonuniform mesh by using the uniform (Formula presented.) bound of the numerical solutions. Finally, some numerical experiments are carried out to validate the theoretical results and the efficiency of the proposed scheme combined with the time adaptive strategy.