Volume 38, Issue 4
High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity

J. Comp. Math., 38 (2020), pp. 580-605.

Published online: 2020-04

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• Abstract

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.

• Keywords

Water waves, Nonlocal viscosity, Finite difference, Spectral method, Convergence order, Decay rate.

35R11, 65L12, 76M22, 65L20

cjxu@xmu.edu.cn (Chuanju Xu)

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@Article{JCM-38-580, author = {Hasan , Mohammad Tanzil and Xu , Chuanju}, title = {High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {4}, pages = {580--605}, abstract = {

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1902-m2017-0280}, url = {http://global-sci.org/intro/article_detail/jcm/16464.html} }
TY - JOUR T1 - High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity AU - Hasan , Mohammad Tanzil AU - Xu , Chuanju JO - Journal of Computational Mathematics VL - 4 SP - 580 EP - 605 PY - 2020 DA - 2020/04 SN - 38 DO - http://doi.org/10.4208/jcm.1902-m2017-0280 UR - https://global-sci.org/intro/article_detail/jcm/16464.html KW - Water waves, Nonlocal viscosity, Finite difference, Spectral method, Convergence order, Decay rate. AB -

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.

Mohammad Tanzil Hasan & Chuanju Xu. (2020). High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity. Journal of Computational Mathematics. 38 (4). 580-605. doi:10.4208/jcm.1902-m2017-0280
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