In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. The memory requirement and computational cost are O ( Q ) and O ( QK ), respectively, where K is the number of the final time step and Q is the number of quadrature points used in the trapezoidal rule. Some numerical experiments are given to confirm our theoretical analysis and the effectiveness of the presented methods.

中文翻译：

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具弱奇异解的非线性时间分数阶动/不动对流扩散方程的误差分析

本文提出了一种加权移位的Grünwald-Letnikov差分（WSGD）勒让德谱方法来求解二维非线性时间分数阶动/不动对流扩散方程。介绍了一种及时处理奇异性的校正方法，并证明了稳定性和收敛性。在数值实现中，基于梯形规则的整体均匀逼近对实线上的积分应用快速方法，以减少内存需求和计算成本。内存需求和计算成本分别为O（Q）和O（QK），其中K是最后时间步的数量，Q是梯形规则中使用的正交点的数量。