Fast, unconditionally energy stable large time stepping method for a new Allen–Cahn type square phase-field crystal model

2019 ◽  
Vol 98 ◽  
pp. 248-255 ◽  
Author(s):  
Fubiao Lin ◽  
Xiaoming He ◽  
Xiaoxia Wen
Keyword(s):  
Phase Field ◽  
Large Time ◽  
Time Stepping ◽  
Crystal Model ◽  
Energy Stable ◽  
Phase Field Crystal

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

2020 ◽  
Author(s):  
Hong-lin Liao ◽  
Bingquan Ji ◽  
Luming Zhang
Keyword(s):  
Error Estimate ◽  
Phase Field ◽  
Implicit Time ◽  
Time Step ◽  
Time Stepping ◽  
Nonlinear Parabolic ◽  
Crystal Model ◽  
Discrete Orthogonal ◽  
Convolution Kernels ◽  
Phase Field Crystal

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 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.

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