Details of Research Outputs

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 := t_{k /} t_{k - 1} < (3 + √17)/2 ≈ 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 rk < 1 + \surd 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.