The mathematical models built based on space-fractional derivatives adequately describe the phenomena incorporating spatial heterogeneity, but their numerical simulation on general domains remains a difficult issue. In this study, we develop a local differential quadrature (DQ) formula based on radial basis functions (RBFs) to discretize Riesz fractional derivatives, which is defined by the functional values at the nodes located in the subdomain around the discrete node. The proposed formula overcomes the drawback of ill-conditioning associated with the global DQ method and enables us to approximate the fractional derivatives on irregular domain with high flexibility and good accuracy. Applying this local DQ formula in space and using weighted average (WA), predictor-corrector (PC) difference schemes in time, we then construct two linearized local RBFs-based DQ schemes for the two-dimensional nonlinear Riesz space-fractional advection-diffusion equations (ADEs). The validity of this local DQ formula is confirmed by numerical results. The feasibility and capability of both local RBFs-based DQ schemes are authenticated by the illustrative tests on some irregular domains and the numerical simulation of Gaussian pulse propagation.

基于空间分数导数建立的数学模型充分描述了包含空间异质性的现象，但它们在一般域上的数值模拟仍然是一个难题。在这项研究中，我们开发了一个基于径向基函数 (RBF) 的局部微分正交 (DQ) 公式来离散 Riesz 分数阶导数，它由位于离散节点周围子域中的节点处的函数值定义。所提出的公式克服了与全局 DQ 方法相关的病态的缺点，使我们能够以高灵活性和良好的精度逼近不规则域上的分数阶导数。在空间中应用这个局部 DQ 公式并在时间上使用加权平均 (WA)、预测器-校正器 (PC) 差异方案，然后，我们为二维非线性 Riesz 空间分数平流扩散方程 (ADE) 构建了两个基于线性化局部 RBF 的 DQ 方案。数值结果证实了该局部 DQ 公式的有效性。通过对一些不规则域的说明性测试和高斯脉冲传播的数值模拟，验证了这两种基于局部 RBF 的 DQ 方案的可行性和能力。