In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has memory requirement and computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations.

中文翻译：

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二维Riesz空间-分数阶扩散方程交替方向隐式有限体积方法的分析和有效实现

在本文中，我们针对二维空间中与时间相关的Riesz空间-分数阶扩散方程，开发了Crank-Nicolson交替方向隐式有限体积方法。给出了基于准则的稳定性和收敛性分析，表明所开发的方法是无条件稳定的，并且在时空上具有二阶精度。此外，我们为数值方案的实现开发了一种无损无矩阵快速共轭梯度方法，该方法仅具有内存需求和每次迭代的计算复杂度，其中N为空间未知数的总数。提出了几个数值实验，以证明所提出的方案用于大规模建模和仿真的有效性和效率。