An adaptive BDF2 implicit time-stepping method for the phase field crystal model

An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal model. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels when the time-step ratios satisfy $r_k:=\tau _k/\tau _{k-1}<3.561$ , which is the zero-stability restriction of the variable-step BDF2 scheme for ordinary differential equations. With the help of discrete orthogonal convolution kernels and corresponding convolution inequalities, an optimal $L^2$ norm error estimate is established under the weak step-ratio restriction $0<r_k<3.561$ to ensure energy stability. As far as we know, this is the first time that such an error estimate is theoretically proved for a nonlinear parabolic equation. Based on tests on random temporal meshes an effective adaptive time-stepping strategy is suggested to efficiently capture the multi-scale behavior and accelerate the numerical simulations.