A novel MBP-preserving and energy-stable BDF2 scheme based on the invariant energy quadratization finite difference method for the Allen-Cahn equation

The Invariant Energy Quadratization (IEQ) framework has emerged as a powerful tool for designing energy-stable numerical schemes for gradient flow problems. However, for the Allen-Cahn equation, it remains an open problem whether the IEQ-type schemes can rigorously preserve the discrete maximum bound principle (MBP)—a critical property ensuring the physical admissibility of solutions. While numerical evidence has consistently suggested that IEQ-based schemes tend to preserve the MBP in practice, no theoretical justification has been established to date. In this work, we develop and analyze a fully discrete, second-order accurate numerical scheme based on the IEQ framework that rigorously satisfies both the discrete energy stability and the MBP. The key innovation lies in a new treatment of the auxiliary variable: instead of using second-order extrapolation, we employ a numerically bounded approximation derived from a first-order MBP-preserving scheme, ensuring that the nonlinear coefficient remains controlled. We prove that the resulting scheme is energy stable, and provably satisfies a discrete MBP under a mild and explicitly characterized time-step condition. In addition, we establish optimal-order error estimates for the fully discrete system. Extensive 2D and 3D numerical experiments are conducted to validate the theoretical analysis and demonstrate the accuracy, efficiency, and structure-preserving properties of the proposed method. To the best of our knowledge, this is the first theoretical result showing that an IEQ-based scheme with second-order time discretization can rigorously preserve the MBP, thereby addressing a fundamental open question in the field.