Implicit integration factor method coupled with Pad\'{e} approximation strategy for nonlocal Allen-Cahn equation

TL;DR

This paper introduces a high-order, energy-stable numerical scheme for the nonlocal Allen-Cahn equation using an implicit integration factor method combined with Padé approximation, achieving high accuracy and stability.

Contribution

It develops a novel high-order numerical method that maintains maximum principle and energy stability for nonlocal Allen-Cahn equations, with proven convergence order.

Findings

The scheme achieves convergence order of O(τ^2 + h^6).

Numerical experiments confirm the method's efficiency and stability.

The fully implicit sixth-order spatial scheme maintains maximum principle and energy stability.

Abstract

The space nonlocal Allen-Cahn equation is a famous example of fractional reaction-diffusion equations. It is also an extension of the classical Allen-Cahn equation, which is widely used in physics to describe the phenomenon of two-phase fluid flows.Due to the nonlocality of the nonlocal operator, numerical solutions to these equations face considerable challenges.It is worth noting that whether we use low-order or high-order numerical differential formulas to approximate the operator, the corresponding matrix is always dense, which implies that the storage space and computational cost required for the former and the latter are the same. However, the higher-order formula can significantly improve the accuracy of the numerical scheme.Therefore, the primary goal of this paper is to construct a high-order numerical formula that approximates the nonlocal operator.To reduce the time step limitation in existing numerical algorithms, we employed a technique combining the compact integration factor method with the Pad\'{e} approximation strategy to discretize the time derivative.A novel high-order numerical scheme, which satisfies both the maximum principle and energy stability for the space nonlocal Allen-Cahn equation, is proposed.Furthermore, we provide a detailed error analysis of the differential scheme, which shows that its convergence order is $\mathcal{O}\left(\tau^2+h^6\right)$.Especially, it is worth mentioning that the fully implicit scheme with sixth-order accuracy in spatial has never been proven to maintain the maximum principle and energy stability before.Finally, some numerical experiments are carried out to demonstrate the efficiency of the proposed method.