A Lagrange Galerkin Scheme for the Numerical Stability of the Shallow Water Equations with the Transmission Boundary Condition through Energy Estimates

This paper presents a study on the shallow water equations (SWEs) with specified Dirichlet and transmission boundary conditions (TBC). A Lagrange-Galerkin (LG) scheme is employed for numerical discretization. The stability of the solution is analyzed using energy estimates. The results shows that the total energy generally decreases over time and that the energy derivatives are non-positive, which confirms the numerical stability of the solution.