The time distributed-order diffusion-wave equation describes radial groundwater flow to or from a well. In the paper, an alternating direction implicit (ADI) Legendre–Laguerre spectral scheme is proposed for the two-dimensional time distributed-order diffusion-wave equation on a semi-infinite domain. The Gauss quadrature formula has a higher computational accuracy than the Composite Trapezoid formula and Composite Simpson formula, which is presented to approximate the distributed order time derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Legendre–Laguerre spectral scheme to avoid introducing the artificial boundary and in time using the weighted and shifted Grünwald–Letnikov difference (WSGD) method. A stability and convergence analysis is performed for the numerical approximation. Some numerical results are illustrated to justify the theoretical analysis.

时间分布阶扩散波方程描述流入或流出井的径向地下水流。在该论文中，针对半无限域上的二维时间分布阶扩散波动方程，提出了一种交替方向隐式（ADI）Legendre-Laguerre谱方案。高斯求积公式比复合梯形公式和复合辛普森公式具有更高的计算精度，该公式用于近似分布阶次时间导数，从而将所考虑的方程转化为多项式分数方程。此外，变换后的方程通过 ADI Legendre-Laguerre 谱方案在空间中离散化以避免引入人为边界，并使用加权和移位 Grünwald-Letnikov 差分 (WSGD) 方法及时求解。对数值近似执行稳定性和收敛性分析。说明了一些数值结果以证明理论分析的合理性。