Details of Research Outputs

A fully implicit two-step backward differentiation formula (BDF2) scheme with variable time steps is considered for solving the phase field crystal (PFC) model by combining with pseudo-spectral method in space. We show that the BDF2 scheme inherits a modified energy dissipation law under a mild ratio restriction A1, i.e., 0 < r: = τ/ τ< r≈ 4.8645 , which justifies the thermodynamic consistency of the PFC model numerically. In addition, the optimal second-order convergence of the proposed scheme is also established under the ratio restriction A1. The proof involves the tools of DOC and DCC kernels, and some generalized properties of the DOC kernels. As far as we know, the ratio restriction A1 required in our results is mildest so far for the variable time-step BDF2 scheme for calculating PFC model. Numerical examples are provided to demonstrate our theoretical results.