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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 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.Comment: 18 page