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arXiv · 2008.00212

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

Hong-lin Liao·Bingquan Ji·Luming Zhang
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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 $r_k:=τ_k/τ_{k-1}<3.561$, 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 $L^2$ norm error estimate is established under the weak step-ratio restriction $0<r_k<3.561$ 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.

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Hong-lin Liao, Bingquan Ji, Luming Zhang. 2020-08-01. An adaptive BDF2 implicit time-stepping method for the phase field crystal model. https://doi.org/10.1093/imanum%2Fdraa075

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