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

Abstract

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmission boundary condition, an inequality for the energy estimate is proved under some assumptions to be satisfied in practical computation. Hence, it is recognized that the transmission boundary condition is reasonable in the sense that the inequality holds true. In the numerical part, based on the theoretical results, the energy estimate of the SWEs with a transmission boundary condition is confirmed numerically by a finite difference method (FDM). The choice of a positive constant c0 used in the transmission boundary condition is investigated additionally. Furthermore, we present numerical results by a Lagrange-Galerkin scheme, which are similar to those by the FDM. From the numerical results, it is found that the transmission boundary condition works well numerically.