Mixed finite element method for 2D Shallow Water Equations with a Relaxation open boundary condition

We study the shallow water equations in a domain with variable bathymetry consisting of a channel and a lagoon. This domain has an open boundary materialized by the opening of the channel on the ocean and is denoted by \Sigma_{\ell}. The flow in the vicinity of this boundary is influenced by the combined effect of the tide and a vortex created by a singular punctual source S\circledast. On \Sigma_{\ell}, we write a relaxation open boundary condition. Using the Crank-Nicholson scheme and the mixed finite element pair P_{1}^{NC}-P_{1}, a completely discrete numerical model is established for which we show the existence and uniqueness of the solution u_{h}, \eta_{h}) in the appropriate space V_{h}\times{Q_{h}}_{/Ker B_{h}^{t}}. This shows that the Relaxation condition leads to a mathematically well-posed problem. Moreover, the comparison of this solution with the experimental results in the Vridi channel proves that the Relaxation condition is physically admissible.