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Volume 10, Issue 2
Numerical Investigation of the Decay Rate of Solutions to Models for Water Waves with Nonlocal Viscosity

S. Dumont & J.-B. Duval

Int. J. Numer. Anal. Mod., 10 (2013), pp. 333-349.

Published online: 2013-10

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  • Abstract

In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

  • Keywords

waterwaves, viscous asymptotical models, long-time asymptotics, fractional derivatives.

  • AMS Subject Headings

35Q35, 35Q53, 76B15, 65M70

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-333, author = {S. Dumont and J.-B. Duval}, title = {Numerical Investigation of the Decay Rate of Solutions to Models for Water Waves with Nonlocal Viscosity}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {2}, pages = {333--349}, abstract = {

In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/571.html} }
TY - JOUR T1 - Numerical Investigation of the Decay Rate of Solutions to Models for Water Waves with Nonlocal Viscosity AU - S. Dumont & J.-B. Duval JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 333 EP - 349 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/571.html KW - waterwaves, viscous asymptotical models, long-time asymptotics, fractional derivatives. AB -

In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

S. Dumont and J.-B. Duval. (2013). Numerical Investigation of the Decay Rate of Solutions to Models for Water Waves with Nonlocal Viscosity. International Journal of Numerical Analysis and Modeling. 10 (2). 333-349. doi:
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