Murshed, Md. Masum; Futai, Kouta; Kimura, Masato; Notsu, Hirofumi Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. (English) Zbl 1462.65110 Discrete Contin. Dyn. Syst., Ser. S 14, No. 3, 1063-1078 (2021). Summary: 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. 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 \(c_0\) 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. The theoretical results along with the numerical results strongly recommend that the transmission boundary condition is suitable for the boundaries in the open sea. Cited in 1 Document MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids 86A05 Hydrology, hydrography, oceanography 35Q35 PDEs in connection with fluid mechanics Keywords:shallow water equations; stability; transmission boundary condition; finite difference method; Lagrange-Galerkin scheme PDFBibTeX XMLCite \textit{Md. M. Murshed} et al., Discrete Contin. Dyn. Syst., Ser. S 14, No. 3, 1063--1078 (2021; Zbl 1462.65110) Full Text: DOI arXiv References: [1] D. Bresch; B. 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