In this paper, we present a rigorous proof of the convergence of first order and second order exponential time differencing (ETD) schemes for solving the nonlocal Cahn-Hilliard (NCH) equation. The spatial discretization employs the Fourier spectral collocation method, while the time discretization is implemented using ETD-based multistep schemes. The absence of a higher-order diffusion term in the NCH equation poses a significant challenge to its convergence analysis. To tackle this, we introduce new error decomposition formulas and employ the higher-order consistency analysis. These techniques enable us to establish the $\ell^\infty$ bound of numerical solutions under some natural constraints. By treating the numerical solution as a perturbation of the exact solution, we derive optimal convergence rates in $\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T; \ell^2)$. We conduct several numerical experiments to validate the accuracy and efficiency of the proposed schemes, including convergence tests and the observation of long-term coarsening dynamics.

翻译：本文对求解非局部Cahn-Hilliard方程的一阶和二阶指数时间差分格式的收敛性给出了严格证明。空间离散采用傅里叶谱配置法，时间离散则通过基于ETD的多步格式实现。NCH方程中高阶扩散项的缺失为其收敛性分析带来了显著挑战。为此，我们引入了新的误差分解公式并采用高阶相容性分析技术。这些方法使我们能够在若干自然约束条件下建立数值解的$\ell^\infty$范数界。通过将数值解视为精确解的扰动，我们推导出了$\ell^\infty(0,T;H_h^{-1})\cap \ell^2(0,T; \ell^2)$空间中的最优收敛速率。我们进行了多项数值实验以验证所提格式的精度与效率，包括收敛性测试以及长期粗化动力学观测。