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

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.

On the Comparison of Evolution Galerkin and Discontinuous Galerkin Schemes

“…Note that we have obtained analogous results of accuracy study in our recent paper on well-balanced schemes, see Ref. 20. Table 6 gives the overview of the computational costs for all of the methods, the DG, FVEG as well as classical FV method.…”

“…14, 15. This research was motivated by the pioneering work of [17][18][19]11,20 a genuinely multidimensional finite volume evolution Galerkin (FVEG) method has been developed, studied extensively from theoretical as well as experimental point of view and applied to various applications. The method is based on the theory of bicharacteristics, which is combined with the finite volume framework.…”

“…We refer the reader to Refs. 18,19,20 for examples of such approximate evolution operators for the Euler equations, shallow water equations as well as for the wave equation system, cf. also Section 4.…”

Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method

“…Lukácová-Medvid'ová et al [3] presented a new wellbalanced finite volume method within the framework of the finite volume evolution Galerkin (FVEG) schemes for the shallow water equations with source terms modeling the bottom topography and the Coriolis forces.…”

On the Stability of Evolution Galerkin Schemes Applied to a Two‐Dimensional Wave Equation System

“…In [11], [15], [6] the FVEG schemes have been generalized to fully nonlinear systems of hyperbolic conservation laws, such as the Euler equations of gas dynamics, shallow water equations as well as the shallow water magnetohydrodynamic equations. For hyperbolic conservation laws with source terms the so-called well-balanced FVEG schemes are proposed in [20]. In general, the FVEG schemes produce very accurate numerical solutions within the CPU time that is comparable to some other well-known finite volume methods.…”