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Volume 17, Issue 3
Splitting ADI Scheme for Fractional Laplacian Wave Equations

Tao Sun & Hai-Wei Sun

DOI: 10.4208/nmtma.OA-2023-0149

Numer. Math. Theor. Meth. Appl., 17 (2024), pp. 697-726.

Published online: 2024-08

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  • Abstract

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.

  • Keywords

Operator splitting, alternative direction implicit scheme, Gohberg-Semencul formula, fractional Laplacian wave equation.

  • AMS Subject Headings

65F05, 65M06, 65M12, 65M15

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-17-697, author = {Sun , Tao and Sun , Hai-Wei}, title = {Splitting ADI Scheme for Fractional Laplacian Wave Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2024}, volume = {17}, number = {3}, pages = {697--726}, abstract = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2023-0149}, url = {http://global-sci.org/intro/article_detail/nmtma/23371.html} }
TY - JOUR T1 - Splitting ADI Scheme for Fractional Laplacian Wave Equations AU - Sun , Tao AU - Sun , Hai-Wei JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 697 EP - 726 PY - 2024 DA - 2024/08 SN - 17 DO - http://doi.org/10.4208/nmtma.OA-2023-0149 UR - https://global-sci.org/intro/article_detail/nmtma/23371.html KW - Operator splitting, alternative direction implicit scheme, Gohberg-Semencul formula, fractional Laplacian wave equation. AB -

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.

Sun , Tao and Sun , Hai-Wei. (2024). Splitting ADI Scheme for Fractional Laplacian Wave Equations. Numerical Mathematics: Theory, Methods and Applications. 17 (3). 697-726. doi:10.4208/nmtma.OA-2023-0149
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