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

Splitting ADI scheme for fractional Laplacian wave equations