Up to eighth-order maximum-principle-preserving methods for the Allen-Cahn equation

In this work, we develop a class of up to eighth-order maximum-principle-preserving (MPP) methods for the Allen-Cahn equation. Beginning with the space-discrete system, we extend the integrating factor two-step Runge-Kutta (IFTSRK) methods and define sufficient conditions for the preservation of the discrete maximum principle. In particular, we combine the IFTSRK methods with the linear stabilization technique to develop the stabilized IFTSRK formulations and successfully derive sufficient conditions to preserve the discrete maximum principle unconditionally. Furthermore, we provide error estimates for these proposed methods. Numerical experiments are carried out to illustrate the high-order accuracy and MPP characteristic of the proposed methods and to verify the efficiency through simulations of the long-time evolutional behavior.