  This study presents a two-step Lagrange-Galerkin scheme for the shallow water
equations with a transmission boundary condition (TBC). Firstly, the
experimental order of convergence of the scheme is shown to see the
second-order accuracy in time. Secondly, the effect of the TBC on a simple
domain is discussed; the artificial reflections are kept from the Dirichlet
boundaries and removed significantly from the transmission boundaries. Thirdly,
the scheme is applied to a complex practical domain, i.e., the Bay of Bengal
region, which is non-convex and includes islands. The effect of the TBC is
discussed again for the complex domain; the artificial reflections are removed
significantly from transmission boundaries, which are set on open sea
boundaries. Based on the numerical results, it is revealed that the scheme has
the following properties; (i) the same advantages of Lagrange-Galerkin methods
(the CFL-free robustness for convection-dominated problems and the symmetry of
the matrices for the system of linear equations); (ii) second-order accuracy in
time; (iii) mass preservation of the function for the water level from the
reference height (until the contact with the transmission boundaries of the
wave); and (iv) no significant artificial reflection from the transmission
boundaries. The numerical results by the scheme are presented in this paper for
the flat bottom topography of the domain. In the next part of this work, Part
II, the scheme will be applied to rapidly varying bottom surfaces and a real
bottom topography of the Bay of Bengal region.
