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 $r_k:={\tau }_k/{\tau }_{k-1}<3.561$, 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.

arXiv comment29 pages, 18 figures, 2 tables

References in corpus (4)

• Modeling Elasticity in Crystal Growth
• Modeling Elastic and Plastic Deformations in Non-Equilibrium Processing Using Phase Field Crystals
• Analysis of adaptive BDF2 scheme for diffusion equations
• An Energy Stable BDF2 Fourier Pseudo-Spectral Numerical Scheme for the Square Phase Field Crystal Equation

Cited by in corpus (8)

• Analysis of adaptive BDF2 scheme for diffusion equations
• Mesh-robustness of an energy stable BDF2 scheme with variable steps for the Cahn-Hilliard model
• Analysis of the second order BDF scheme with variable steps for the molecular beam epitaxial model without slope selection
• Stability of variable-step BDF2 and BDF3 methods
• Compatible $L^2$ norm convergence of variable-step L1 scheme for the time-fractional MBE mobel with slope selection
• $L^2$ norm error estimates of BDF methods up to fifth-order for the phase field crystal model
• An efficient two-grid fourth-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation
• Energy dissipation law and maximum bound principle-preserving linear BDF2 schemes with variable steps for the Allen-Cahn equation