A Maximum Bound Principle-Preserving Integrating Factor BDF2 Scheme with Its Linear Iteration Algorithm for the Allen-Cahn Equation

Abstract

The maximum bound principle (MBP) is an essential tool in understanding the key physical properties of parabolic partial differential equations. In this paper, we develop and analyze a novel, MBP-preserving, second-order nonuniform scheme for the Allen-Cahn model with general potential, namely, the integrating factor BDF2 scheme. Specifically, we employ the MBP-preserving iteration and an enhanced kernel recombination technique to show that the integrating factor BDF2 scheme preserves the MBP under a mild time-step condition and a time-step ratio condition. It is worth noting that the integrating factor approach enables the time-step condition to be decoupled from the spatial step size, thus circumventing the stringent parabolic Courant-Friedrichs-Lewy condition $τ = \mathcal{O}(h^2)$. Additionally, the MBP-preserving iteration technique leads to an efficient and convergent linear iterative algorithm for this nonlinear scheme. We provide error estimates in the maximum norm for nonuniform time meshes without requiring global Lipschitz continuity of the potential function. Lastly, we conduct thorough numerical experiments to verify the efficacy and performance of the proposed scheme.