A new high-order maximum-principle-preserving explicit Runge–Kutta method for the nonlocal Allen–Cahn equation

Abstract

We develop the high-order maximum principle preserving schemes based on combining the scalar variable with the explicit strong stability preserving Runge–Kutta (eSSPRK) methods, which can be applied to a class of gradient flows. In this work, we use the methods to deal with the nonlocal Allen–Cahn equation. There are two remarkable properties of the scheme. On the one hand, by using the introduced scalar variable to control the nonlinear term, and adopting the eSSPRK method to temporal discretization, we only need to solve the linear equations with constant coefficients at each time level. On the other hand, we introduce the first-order approximation of the energy balance equation instead of the dynamic equation for the scalar variable, and we demonstrate that the proposed schemes are unconditionally energy stable. In particular, the discrete schemes are verified to preserve the maximum bound principle under the time-step restriction. Moreover, error estimates and the asymptotic compatibility of the proposed methods are proved, respectively. Numerical examples are performed to illustrate the efficiency and accuracy of the numerical schemes.