The non-hydrostatic depth-integrated model we developed to study solitary waves passing undisturbed in shape through a porous structure, involves hydrodynamic pressure. The equations are nonlinear, diffusive, and weakly dispersive wave equation for describing solitary wave propagation in a porous medium. We solve the equation numerically using a staggered finite volume with a predictor-corrector method. To demonstrate our non-hydrostatic scheme’s performance, we implement our scheme for simulating solitary waves over a flat bottom in a free region to examine the balance between dispersion and nonlinearity. Our computed waves travel undisturbed in shape as expected. Furthermore, the numerical scheme is used to simulate the solitary waves pass through a porous structure. Results agree well with results of a central finite difference method in space and a fourth-order Runge-Kutta integration technique in time for the Boussinesq model. When we quantitatively compare the wave amplitude reduction from our numerical results to experimental data, we find satisfactory agreement for the wave transmission coefficient.

1. [1] K. Kathiresan and N. Rajendran, “Coastal mangrove forests mitigated tsunami,” Estuarine, Coastal and Shelf Science, Vol.65, No.3, pp. 601–606, 2005.
2. [2] F. Dahdouh-Guebas, “Mangrove forests and tsunami protection,” McGraw-Hill Yearbook of Science & Technology, pp. 187–191, 2006.
3. [3] S. Pudjaprasetya and I. Magdalena, “Wave Energy Dissipation over Porous Media,” Applied Mathematical Sciences, Vol.7, No.59, pp. 2925–2937, 2013.
4. [4] Z. Gu and H. Wang, “Gravity Waves over Porous Bottoms,” Coastal Engineering, Vol.15, pp. 497–524, 1991.
5. [5] I. Magdalena, S. R. Pudjaprasetya, and L. H. Wiryanto, “Wave Interaction with an Emerged Porous Media,” Advances in Applied Mathematics and Mechanics, Vol.6, No.5, pp. 680–692, Jun., 2015.
6. [6] P. L.-F. Liu and J. Wen, “Nonlinear diffusive surface waves in porous media,” Journal of Fluid Mechanics, Vol.347, pp. 119–139, Sep., 1997.
7. [7] P. J. Lynett, P. L.-F. Liu, I. J. Losada, and C. Vidal, “Solitary Wave Interaction with Porous Breakwaters,” Journal of Waterway, Port, Coastal, and Ocean Engineering, Vol.126, No.6, pp. 314–322, 2000.
8. [8] C. Vidal, M. A. Losada, R. Medina, and J. Rubio, “Solitary Wave Transmission through Porous Breakwaters,” Coastal Engineering, pp. 1073–1083, 1988.
9. [9] G. Stelling and S. Duinmeijer, “A staggered conservative scheme for every Froude number in rapidly varied shallow water flows,” International Journal for Numerical Methods in Fluids, Vol.43, pp. 1329–1354, 2003.
10. [10] S. Pudjaprasetya and I. Magdalena, “Momentum Conservative Scheme for Dam break and Wave Run up Simulations,” East Asia Journal on Applied Mathematics, Vol.4, No.2, pp. 152–165, 2014.
11. [11] I. Magdalena, N. Erwina, and S. Pudjaprasetya, “Staggered Momentum Conservative Scheme For Radial Dam Break Simulation,” J. Sci. Comput, Vol.3, No.65, pp. 867–874, 2015.
12. [12] F. Engelund, “On the laminar and turbulent flows of ground water through homogeneous sand,” Transaction of The Danish Academy of Technical Sciences, Vol.3, No.4, 1953.