Volume 7, Issue 4

East Asian J. Appl. Math., 7 (2017), pp. 643-657.

Published online: 2018-02

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• Abstract

In this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

65M10, 78A48

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@Article{EAJAM-7-643, author = {}, title = {Three-Layer Non-Hydrostatic Staggered Scheme for Free Surface Flow}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {7}, number = {4}, pages = {643--657}, abstract = {

In this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.171016.300517a}, url = {http://global-sci.org/intro/article_detail/eajam/10711.html} }
TY - JOUR T1 - Three-Layer Non-Hydrostatic Staggered Scheme for Free Surface Flow JO - East Asian Journal on Applied Mathematics VL - 4 SP - 643 EP - 657 PY - 2018 DA - 2018/02 SN - 7 DO - http://doi.org/10.4208/eajam.171016.300517a UR - https://global-sci.org/intro/article_detail/eajam/10711.html KW - The 2D Euler equations, non-hydrostatic scheme, solitary wave, undular bore. AB -

In this paper, a finite difference algorithm using a three-layer approximation for the vertical flow region to solve the 2D Euler equations is considered. In this algorithm, the pressure is split into hydrostatic and hydrodynamic parts, and the predictor-corrector procedure is applied. In the predictor step, the momentum hydrostatic model is formulated. In the corrector step, the hydrodynamic pressure is accommodated after solving the Laplace equation using the Successive Over Relaxation (SOR) iteration method. The resulting algorithm is first tested to simulate a standing wave over an intermediate constant depth. Dispersion relation of the scheme is derived, and it is shown to agree with the analytical dispersion relation for kd < π with 94% accuracy. The second test case is a solitary wave simulation. Our computed solitary wave propagates with constant velocity, undisturbed in shape, and confirm the analytical solitary wave. Finally, the scheme is tested to simulate the appearance of the undular bore. The result shows a good agreement with the result from the finite volume scheme for the Boussinesq-type model by Soares-Frazão and Guinot (2008).

Ade C. Bayu, S. R. Pudjaprasetya & I. Magdalena. (2020). Three-Layer Non-Hydrostatic Staggered Scheme for Free Surface Flow. East Asian Journal on Applied Mathematics. 7 (4). 643-657. doi:10.4208/eajam.171016.300517a
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