In this paper, we investigate the numerical solution of the two-dimensional fractional Laplacian wave equations. After splitting out the Riesz fractional derivatives from the fractional Laplacian, we treat the Riesz fractional derivatives with an implicit scheme while solving the rest part explicitly. Thanks to the tensor structure of the Riesz fractional derivatives, a splitting alternative direction implicit (S-ADI) scheme is proposed by incorporating an ADI remainder. Then the Gohberg-Semencul formula, combined with fast Fourier transform, is proposed to solve the derived Toeplitz linear systems at each time integration. Theoretically, we demonstrate that the S-ADI scheme is unconditionally stable and possesses second-order accuracy. Finally, numerical experiments are performed to demonstrate the accuracy and efficiency of the S-ADI scheme.

翻译：本文研究二维分数阶拉普拉斯波动方程的数值求解方法。通过从分数阶拉普拉斯算子中分离出Riesz分数阶导数，我们对Riesz分数阶导数采用隐式格式处理，同时显式求解其余部分。利用Riesz分数阶导数的张量结构，引入ADI余项构造了一种分裂交替方向隐式（S-ADI）格式。进而提出结合快速傅里叶变换的Gohberg-Semencul公式，用于求解每个时间积分步中生成的Toeplitz线性系统。理论分析表明，S-ADI格式具有无条件稳定性和二阶精度。最后通过数值实验验证了S-ADI格式的精确性与高效性。