Abstract Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.

中文翻译：

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二维多项时空分数阶非线性扩散波方程的一种数值方法

摘要 最近，已经开发了许多用于求解单项时空分数扩散波方程的数值方案。其中，由于时间规律性低的解决方案，一些流行的方法是通过使用分级网格来构建的。在本文中，我们为二维多项时空分数非线性扩散波方程提出了一种有效的交替方向隐式（ADI）方案。首先，将所考虑的问题与黎曼-刘维尔积分和多项卡普托导数等价转化为其偏积分微分形式。其次，ADI方案是通过使用一阶近似和L1近似在时间上逼近项，并使用分数中心差分在空间上离散多项Riesz分数导数来构建的。此外，通过 Caputo 导数和 Riemann-Liouville 积分的指数和技术讨论了所提出的 ADI 方案的快速实现。然后，严格建立所提出的ADI方案的可解性、无条件稳定性和收敛性。最后，给出了两个数值例子来支持我们的理论结果，并证明了快速 ADI 方案的计算性能。