In this study, we develop a second-order finite difference scheme based on the shifted convolution quadrature (SCQ) framework that approximates the space-fractional derivatives at a shifted node $x_{n-\theta }$ where $\theta$ may not necessarily be an integer. By applying the proposed method for a space-fractional advection–diffusion equation in the spacial direction and the Crank–Nicolson scheme for the time variable discretization, we analyze the von Neumann stability for the fully discrete scheme. Further, we explore the impact of different $\theta$ on the robustness of our scheme for weak regular solutions and compare that with the shifted Grünwald–Letnikov formula. The results confirm the necessity of introducing non-integer shifted parameters $\theta$.

在这项研究中，我们基于位移卷积正交（SCQ）框架开发了一种二阶有限差分方案，该方案近似于位移节点处的空间分数导数 $X_{ñ-\theta }$ 哪里 $\theta$不一定是整数。通过将所提出的方法用于空间方向上的空间分数对流扩散方程和时变离散化的Crank-Nicolson方案，我们分析了完全离散方案的von Neumann稳定性。此外，我们探索了不同的影响$\theta$对我们弱弱正则解方案的鲁棒性进行了比较，并将其与移位的Grünwald–Letnikov公式进行了比较。结果证实了引入非整数移位参数的必要性$\theta$。