On energy stable, maximum-principle preserving, second order BDF scheme with variable steps for the Allen-Cahn equation

Abstract

In this work, we investigate the two-step backward differentiation formula (BDF2) with nonuniform grids for the Allen-Cahn equation. We show that the nonuniform BDF2 scheme is energy stable under the time-step ratio restriction $r_k:={\tau }_k/{\tau }_{k-1}<(3+\sqrt{17})/2\approx \mathrm{3.561.}$ Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum principle of BDF2 scheme under the time-step ratio restriction $r_k<1+\sqrt{2}\approx 2.414$ and a practical time step constraint. The second-order rate of convergence in the maximum norm is also presented. Numerical experiments are provided to support the theoretical findings.

24 pages, 1 table and 25 figures