SINGLE-LAYER FINITE ELEMENT APPROXIMATION OF 3D NAVIER-STOKES EQUATIONS WITH FREE BOUNDARIES FOR SHALLOW WATER FLOWS

Abstract

In order to simulate flows in the shallow water limit, the full incompressible Navier–Stokes equations with free boundaries are solved using a single layer of finite elements. This implies a polynomial approximation of the velocity profile in the vertical direction, which in turn distorts the wave speed. This fact is verified by numerical results: the wave speed depends on the vertical discretization. When at least two layers of finite elements are used, the boundary layer at the bottom can be simulated and the correct solution for the shallow water limit is recovered. Then this algorithm is applied to the prediction of Tsunami event.