A maximum bound principle-preserving, second-order BDF scheme with variable steps for the generalized Allen–Cahn equation

The maximum bound principle (MBP) is a crucial characteristic for a board class of semilinear parabolic equations, making it essential to preserve this feature in numerical simulations, particularly for problems involving degenerate mobility and logarithmic potential function. In this paper, we focus on the nonuniform second-order backward differentiation formula (BDF2) for the Allen–Cahn (AC) model with general potential and variable mobility. With moderate restrictions on the time step and a condition on the time-step ratio, the nonuniform BDF2 scheme has been shown to satisfy the discrete MBP. The newly improved kernels recombination technique and MBP-preserving iteration technique significantly contribute to this analysis. This work represents the first result of a nonuniform BDF2 scheme that preserves the MBP for AC-type equations with a logarithmic potential function. With the discrete MBP, the maximum norm convergence analysis is obtained without requiring any Lipschitz conditions on the nonlinear bulk force. Additionally, various numerical experiments are conducted for the generalized AC model, incorporating an adaptive time-stepping algorithm.