Variable Step-Size BDF3 Method for Allen-Cahn Equation (Journal of Computational Mathematics, 2024)
In this work, we analyze the three-step backward differentiation formula (BDF3) method for solving the Allen-Cahn equation on variable grids. For BDF2 method, the discrete orthogonal convolution (DOC) kernels are positive, the stability and convergenceanalysis are well established in [Liao and Zhang, Math. Comp., 90 (2021), 1207–1226]and [Chen, Yu, and Zhang, arXiv:2108.02910, 2021]. However, the numerical analysis forBDF3 method with variable steps seems to be highly nontrivial due to the additional degrees of freedom and the non-positivity of DOC kernels. By developing a novel spectralnorm inequality, the unconditional stability and convergence are rigorously proved underthe updated step ratio restriction $r_k :=\tau_k/\tau_{k−1}≤1.405$ for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best ofour knowledge, this is the first theoretical analysis of variable steps BDF3 method for theAllen-Cahn equation.
Keywords: Variable step-size BDF3 method; Allen-Cahn equation; Spectral norm inequality; Stability and convergence analysis.