A two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition and its application to the Bay of Bengal region

This study presents a two-step Lagrange-Galerkin scheme for the shallow water equations with a transmission boundary condition (TBC), which is of second order in time and maintains the two advantages of the Lagrange-Galerkin methods, i.e., the CFL-free robustness for convection-dominated problems and the symmetry of the resulting coefficient matrices for the system of linear equations. The two material derivatives in non-conservative and conservative forms are discretized based on the ideas of the two-step backward difference formula of degree two along the trajectory of the fluid particle. Numerical results by the scheme are presented. 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 and the symmetry of the matrices); (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.