Well-balanced finite volume evolution Galerkin methods for the shallow water equations
This work provides a well-balanced numerical scheme for geophysical flow simulations, addressing the need for accurate preservation of steady states in shallow water models.
The authors present a well-balanced finite volume evolution Galerkin (FVEG) method for shallow water equations with bottom topography and Coriolis forces, proving well-balancing for stationary states and steady jets, and confirming reliability through numerical experiments.
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.