An adaptive BDF2 implicit time-stepping method for the phase field crystal model
Abstract
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 if the time-step ratios rk:=τk/τk-1<3.561, a recent zero-stability restriction of variable-step BDF2 scheme for ordinary differential problems. By using the discrete orthogonal convolution kernels and the corresponding convolution inequalities, an optimal L2 norm error estimate is established under the weak step-ratio restriction 0<rk<3.561 ensuring the energy stability. This is the first time such error estimate is theoretically proved for a nonlinear parabolic equation. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is suggested to efficiently capture the multi-scale behaviors and to accelerate the numerical simulations.