A Two-dimensional Non-hydrostatic Finite Element Model for Wave Propagation

Surface waters are normally modeled using depth-integrated models for good accuracy and efficiency. Due to the shallow water condition, the simulated flow is assumed to be driven by the hydrostatic pressure. When wave dynamics is concerned, the non-hydrostatic pressure has been found to be indispensable for producing accurate results. Implementing such a modeling capability in a general depth-integrated two-dimensional model would benefit coastal wave study and applications. In this paper, a two-dimensional finite element model for simulating wave propagation is presented. The conservation form of shallow water equations with extra non-hydrostatic pressure terms and a depth-integrated vertical momentum equation are solved based on an existing shallow water flow solver, CCHE2D. A conservation scheme for the advection terms is developed to ensure that the momentum is conserved at the discretized level. Overall, the governing equations are solved semi-implicitly: the shallow water equations are firstly solved for the provisional velocity, and then the non-hydrostatic pressure, which is formulated by the continuity equation to achieve a divergence-free velocity field, is solved implicitly and immediately it is used to correct the provisional velocity, finally the depth-integrated continuity equation is solved for the free surface elevation to satisfy global mass conservation. The wave model is verified and validated by several benchmark cases, and the results show that it can properly simulate weakly dispersive waves propagation, wave shoaling, diffraction, refraction and focusing.