Notes on Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions

TL;DR

This paper analyzes Galerkin-finite element methods for the Shallow Water equations with characteristic boundary conditions, providing error estimates and demonstrating excellent wave absorption properties through numerical experiments.

Contribution

It offers the first rigorous error analysis for finite element methods applied to the Shallow Water equations with transparent boundary conditions.

Findings

Error estimates for semi-discrete Galerkin methods

Fourth-order Runge-Kutta scheme maintains absorption properties

Numerical experiments confirm minimal wave reflection

Abstract

We consider the Shallow Water equations in the supercritical and subcritical cases in one space variable,posed in a finite spatial interval with characteristic boundary conditions at the endpoints, which, as is well known, are transparent,i.e. allow outgoing waves to exit without generating spurious reflected waves. Assuming that the resulting initial-boundary-value problems have smooth solutions,we approximate them in space using standard Galerkin-finite element methods and prove L^2 error estimates for the semidiscrete problems on quasiuniform meshes.We discretize the problems in the temporal variable using an explicit,fourth-order accurate Runge-Kutta scheme and check, by means of numerical experiment, that the resulting fully discrete schemes have excellent absorption properties.