Maximum bound principle preserving additive partitioned Runge-Kutta schemes for the Allen-Cahn equation

In this paper, a novel framework for constructing the temporal high-order numerical schemes that inherit the maximum bound principle (MBP) of the Allen-Cahn equation is proposed. The original Allen-Cahn equation is converted into an equivalent system by introducing an auxiliary variable. Based on the new system, we put forward and analyze high-order (up to the fourth-order) additive partitioned Runge-Kutta (APRK) schemes for the time integration of the Allen-Cahn equation, which satisfies the discrete MBP under reasonable time step constraints. A simple sufficient condition is also given to ensure that a class of APRK methods preserves the discrete MBP and that the resulting scheme is linearly implicit. The convergence order in the discrete L∞-norm is further provided by utilizing the established discrete MBP, which plays a vital role in the analysis. Several numerical experiments are carried out to validate our theoretical results.