Well-balanced finite volume evolution Galerkin methods for the shallow water equations

TL;DR

This paper introduces a novel well-balanced finite volume evolution Galerkin (FVEG) method for the shallow water equations, effectively capturing steady states and complex wave phenomena with proven stability and accuracy.

Contribution

The paper develops a new well-balanced FVEG scheme that explicitly accounts for wave directions, improving the numerical treatment of shallow water equations with source terms.

Findings

Successfully preserves steady states and steady jets.

Numerical experiments confirm the scheme's reliability and accuracy.

Method generalizes to complex systems of balance laws.

Abstract

We present a new well-balanced finite volume method within the framework of the finite volume evolution Galerkin (FVEG) schemes. The methodology will be illustrated for the shallow water equations with source terms modelling the bottom topography and Coriolis forces. Results can be generalized to more complex systems of balance laws. The FVEG methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We derive a well- balanced approximation of the integral equations and prove that the FVEG scheme is well-balanced for the stationary steady states as well as for the steady jets in the rotational frame. Several numerical experiments for stationary and quasi-stationary states as well as for steady jets confirm the reliability of the well-balanced FVEG scheme.