Discontinuous Galerkin well-balanced schemes using augmented Riemann solvers with application to the shallow water equations

Abstract

Abstract High order methods are becoming increasingly popular in shallow water flow modeling motivated by their high computational efficiency (i.e. the ratio between accuracy and computational cost). In particular, Discontinuous Galerkin (DG) schemes are very well suited for the resolution of the shallow water equations (SWE) and related models, being a competitive alternative to the traditional finite volume schemes. In this work, a novel framework for the construction of DG schemes using augmented Riemann solvers is proposed. Such solvers incorporate the source term at cell interfaces in the definition of the Riemann problem, allowing definition of two different inner states in the so-called star region. The benefits of this family of solvers lie in the exact preservation of the Rankine–Hugoniot condition at cell interfaces at the discrete level, ensuring the preservation of equilibrium solutions (i.e. the well-balanced property) without requiring extra corrections of the numerical fluxes. The semi-discrete DG operator becomes nil automatically under equilibrium conditions, provided the use of suitable quadrature rules. The proposed scheme is applied to the Burgers' equation with geometric source term and to the SWE. The numerical results evidence that the proposed scheme achieves the prescribed convergence rates and preserves the equilibrium states of relevance with machine precision.