Implicit integration factor method coupled with Padé approximation strategy for nonlocal Allen-Cahn equation

TL;DR

The paper addresses numerical solution of the space nonlocal Allen–Cahn equation with a Riesz derivative $\mathcal{L}^\gamma$ for $\gamma\in(1,2]$, proposing a high‑order, structure‑preserving scheme. It combines a sixth‑order spatial discretization of the nonlocal operator, derived from a new generating function $G(w)$ with coefficients $g_m^{(\gamma)}$, with an implicit integration factor time discretization using a [1,1] Padé approximation and trapezoidal quadrature. The authors establish unique solvability, a discrete maximum principle with time step $0<\tau\le 1$ independent of $\epsilon$ and $h$, discrete energy stability under tau bounds linked to $g_0^{(\gamma)}$ and $g_2^{(\gamma)}$, and an $L^\infty$ convergence rate of $\mathcal{O}(\tau^2+h^6)$. Numerical experiments corroborate the theoretical results, demonstrating high accuracy and robust stability, with improved time-step flexibility and efficiency over prior approaches.

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 Padé 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 $\mathcal{O}\left(τ^2+h^6\right)$.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.

Figures (6)

• Figure 1: Evolutions of the error surface for $\gamma=1.3$ and $\epsilon=0.005$, where $\tau=0.005$ and $h=0.005$.
• Figure 2: Evolutions of the error surface for $\gamma=1.7$ and $\epsilon=0.01$, where $\tau=0.02$ and $h=0.001$.
• Figure 3: Evolutions of the maximum norm of the numerical solutions with different $\gamma$ values for $\tau=1$.
• Figure 4: Evolutions of the maximum norm of the numerical solutions with different $\gamma$ values for $\tau=0.5$.
• Figure 5: Evolutions of the maximum norm of the numerical solutions with different $\gamma$ values for $\tau=0.1$.

Theorems & Definitions (31)

• Theorem 2.1
• Proof 1
• Lemma 2.2
• Proof 2
• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• Proof 3
• Lemma 3.4
• Proof 4