In this paper, we present a robust and fully discretized method for solving the time fractional diffusion equation with high-contrast multiscale coefficients. We establish the homogenized equation in a coarse mesh using a multicontinuum approach and employ the exponential integrator method for time discretization. The multicontinuum upscaled model captures the physical characteristics of the solution for the high-contrast multiscale problem, including averages and gradient effects in each continuum at the coarse scale. We use the exponential integration method to address the nonlocality induced by the time fractional derivative and the stiffness from the multiscale coefficients in the semi-discretized problem. Convergence analysis of the numerical scheme is provided, along with illustrative numerical examples. Our results demonstrate the accuracy, efficiency, and improved stability for varying order of fractional derivatives.

翻译：本文提出了一种鲁棒且完全离散化的方法，用于求解具有高对比度多尺度系数的时间分数阶扩散方程。我们采用多连续介质方法在粗网格上建立均匀化方程，并利用指数积分器方法进行时间离散化。该多连续介质升尺度模型能够捕捉高对比度多尺度问题解的物理特性，包括粗尺度上各连续介质内的平均值和梯度效应。我们采用指数积分方法来处理半离散问题中由时间分数阶导数引起的非局部性以及多尺度系数带来的刚度。本文提供了数值格式的收敛性分析，并辅以说明性的数值算例。我们的结果表明，该方法在不同分数阶导数阶数下均具有准确性、高效性以及提升的稳定性。