Backward differentiation formula

Quantum mechanics

Discretization

Norm (philosophy)

Pure mathematics

Statistics

Differential equation

Applied mathematics

Field (mathematics)

Differential algebraic equation

Computer science

Time stepping

Dissipation

Thermodynamics

Physics

Nonlinear system

Mathematics

Political science

Discrete time and continuous time

Mathematical analysis

Geometry

Ordinary differential equation

Polygon mesh

Artificial neural network

Machine learning

Convolution (computer science)

IMA Journal of Numerical Analysis

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.

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