Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In theoretical part, using suitable energy, we begin deriving an equality which implies energy estimate SWEs Dirichlet slip conditions. For condition, inequality for is proved under some assumptions to be satisfied in practical computation. numerical based on results, confirmed numerically by finite difference method (FDM). The choice positive constant \begin{document}$ c_0 $\end{document} used investigated additionally. Furthermore, present results Lagrange-Galerkin scheme, similar those FDM. along strongly recommend that boundaries open sea.