Abstract In this paper, we propose a fast compact implicit integration factor (FcIIF) method with non-uniform time meshes for solving the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation. The weighted and shifted Gruwald-Letnikov (WSGD) approximation is employed to the spatial discretization of the equation, and a system of nonlinear ordinary differential equations (ODEs) in matrix form is obtained. Since the cIIF method can provide excellent stability properties with good efficiency by decoupling the treatment of the diffusion and reaction terms, a fast cIIF (FcIIF) method with non-uniform time meshes is developed to solve the resultant nonlinear system of ODEs. Compared with the cIIF method, the proposed FcIIF method avoids the direct calculation of dense exponential matrices and requires less computational cost. The stability, accuracy and effectiveness of the proposed method are verified by the linear stability analysis and various numerical experiments.

中文翻译：

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二维非线性 Riesz 空间分数反应扩散方程的非均匀网格快速紧凑隐式积分因子法

摘要 在本文中，我们提出了一种具有非均匀时间网格的快速紧凑隐式积分因子（FcIIF）方法，用于求解二维非线性 Riesz 空间分数反应扩散方程。将加权和移位的 Gruwald-Letnikov (WSGD) 近似用于方程的空间离散化，并获得矩阵形式的非线性常微分方程 (ODE) 系统。由于 cIIF 方法可以通过解耦扩散项和反应项的处理来提供出色的稳定性和良好的效率，因此开发了一种具有非均匀时间网格的快速 cIIF (FcIIF) 方法来求解合成的 ODE 非线性系统。与cIIF方法相比，所提出的FcIIF方法避免了密集指数矩阵的直接计算，并且需要更少的计算成本。