Variable step-size BDF3 method for Allen-Cahn equation

TL;DR

This paper provides the first rigorous stability and convergence analysis of the variable step-size BDF3 method for the Allen-Cahn equation, expanding understanding of its numerical behavior on non-uniform grids.

Contribution

The paper develops a novel spectral norm inequality to prove unconditional stability and convergence of the BDF3 method with variable steps for the Allen-Cahn equation.

Findings

Unconditional stability and convergence are established for BDF3 with step ratio up to 1.405.

Numerical experiments confirm the theoretical results.

The analysis extends the understanding of variable step BDF methods for nonlinear PDEs.

Abstract

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 convergence analysis are well established in [Liao and Zhang, \newblock Math. Comp., \textbf{90} (2021) 1207--1226; Chen, Yu, and Zhang, \newblock SIAM J. Numer. Anal., Major Revised]. However, the numerical analysis for BDF3 method with variable steps seems to be highly nontrivial, since the DOC kernels are not always positive. By developing a novel spectral norm inequality, the unconditional stability and convergence are rigorously proved under the updated step ratio restriction $r_k:=\tau_k/\tau_{k-1}\leq 1.405$ (compared with $r_k\leq 1.199$ in [Calvo and Grigorieff, \newblock BIT. \textbf{42} (2002) 689--701]) for BDF3 method. Finally, numerical experiments are performed to illustrate the theoretical results. To the best of our knowledge, this is the first theoretical analysis of variable steps BDF3 method for the Allen-Cahn equation.