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.

翻译：本研究提出了一种适用于带传输边界条件（TBC）的浅水方程的两步拉格朗日-伽辽金格式。首先，通过数值实验验证了该格式的收敛阶，证明了其在时间上具有二阶精度。其次，在简单计算域中讨论了TBC的效果：人工反射被有效抑制于Dirichlet边界之外，并在传输边界处被显著消除。随后，将该格式应用于一个复杂的实际计算域——孟加拉湾区域（该区域为非凸区域且包含岛屿），再次考察TBC的效果；结果表明，在设定于开阔海域边界的传输边界处，人工反射被大幅消除。基于数值结果，揭示了该格式具有以下特性：(i) 兼具拉格朗日-伽辽金方法的优势（对流主导问题中无需考虑CFL条件的鲁棒性，以及线性方程组矩阵的对称性）；(ii) 时间二阶精度；(iii) 水位函数相对于参考高度的质量守恒性（直至波浪与传输边界接触）；(iv) 传输边界处无显著人工反射。本文给出了该格式在平坦底床地形下的数值计算结果。作为本研究的后续部分，第二部分将把该格式应用于快速变化的底床表面及孟加拉湾区域的实际底床地形。