A Second-Order Maximum Bound Principle-Preserving Implicit-Explicit Exponential Runge–Kutta Scheme for the Allen–Cahn Equation with a General Mobility

We develop a novel two-stage second-order coupled implicit-explicit exponential Runge–Kutta (IMEX-ERK2) scheme for the Allen–Cahn (AC) equation with a general mobility (GAC). This scheme utilizes a second-order central difference scheme for the spatial discretization of the diffusion term, combined with a linear stabilization term to ensure the stability of the numerical scheme. The proposed scheme is distinguished by its ability to unconditionally preserve the maximum bound principle (MBP), even in the presence of variable coefficients. The second-order convergence rate in the discrete L^∞-norm sense is rigorously proved. Additionally, the energy stability is analyzed for the classical constant mobility case. Comprehensive numerical experiments are performed to validate the performance of the scheme. Numerical results reveal a significant reduction in computational complexity compared with existing second-order exponential time differencing Runge–Kutta (ETDRK) methods, thereby enhancing computational efficiency without sacrificing accuracy.