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. Part I: Flat bottom topography

TL;DR

This paper introduces a two-step Lagrange-Galerkin scheme for shallow water equations with transmission boundary conditions, demonstrating second-order accuracy, reduced artificial reflections, and applicability to complex regions like the Bay of Bengal.

Contribution

The paper develops a novel two-step Lagrange-Galerkin scheme with transmission boundary conditions for shallow water equations, showing improved accuracy and boundary reflection handling.

Findings

Second-order accuracy in time confirmed.

Significant reduction of artificial reflections at boundaries.

Effective application to complex, real-world domain of the Bay of Bengal.

Abstract

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.