ON ENERGY STABLE, MAXIMUM-PRINCIPLE PRESERVING, SECOND-ORDER BDF SCHEME WITH VARIABLE STEPS FOR THE ALLEN-CAHN EQUATION

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 + root 17)/2 approximate to 3.561. Moreover, by developing a novel kernel recombination and complementary technique, we show, for the first time, the discrete maximum bound principle of the BDF2 scheme under the time-step ratio restriction r(k) < 1 + root 2 approximate to 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.