A Class of High Order Unconditional Maximum Principle Preserving Extended IFRK Schemes for Allen-Cahn Equation

Source abstract

Abstract. This paper proposes a novel framework for constructing up to sixth-order, explicit, and unconditional maximum principle preserving extended integrating factor Runge–Kutta (eIFRK) schemes for the Allen-Cahn equation. The proposed framework significantly relaxes the constraints on the exponential coefficients and offers a recursive procedure to establish high-order schemes. Satisfying the requirement of the local truncation error in the numerical integration, derived from Duhamel’s formula, provides an innovative approach to achieve high-order accuracy instead of solving the order conditions directly. Specifically, in order to derive the target order scheme, Gauss quadrature is applied to approximate the implicit integration, where one order lower scheme is employed to compute explicitly at quadrature points. Consequently, maximum bound principle and [Formula: see text] norm analysis are uniformly proved for eIFRK4–6. Numerical experiments on eIFRK schemes validate their accuracy, maximum-norm preservation, and energy stability. In comparison to the classic second- and third-order sIFRK methods [SIAM J. Sci. Comput., Vol 43, 2021, pp. A1780-A1802], the presented high-order eIFRK schemes, particularly eIFRK6, exhibit enhanced efficiency and stability during long-term simulations, especially for large time steps.