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

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 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.

Regularized Approach for Bingham Viscoplastic Shallow Flow Using the Discontinuous Galerkin Method

“…In this section, we detail the derivatives for the residual vectors of Equation (49). The derivatives are essential to building the tangent terms for our Newton System (50).…”

“…Numerous applications of this method have been developed, and many textbooks offer different introductions to this active research area [33]. Researchers have successfully used the discontinuous Galerkin method to solve the shallow-water equations, e.g., [56,46,2,31,25,19,20,49]. Also, many studies included the treatment of wet/dry fronts, e.g., [9,24,13,60,42,30,14].…”

A POD-based ROM strategy for the prediction in time of advection-dominated problems

“…where ũ(x, t) = u(x, t). The CT-FOM of the 1D inviscid Burgers' equation is obtained by means of the FV method [28,30,33] ũn+1…”

“…The 1D CT-ROM strategy is used to predict the evolution in time of the 2D hyperbolic problem (30). To do this, first, the IC is transformed from the physical space into the Radon domain, i.e., the (s, α) domain.…”

Development of Pod-Based Reduced Order Models Applied to Shallow Water Equations Using Augmented Riemann Solvers

“…In order to ensure stable, robust and accurate solutions in presence of bed variations and friction terms over dry beds, it is necessary to take into account numerical corrections, such as well-balancing, that keeps the discrete equilibrium with machine precision [4,5,19]. Augmented Riemann solvers are designed to preserve equilibrium in presence of source terms [20]. In addition to well-balancing, it is well known that other numerical corrections are necessary to fix some unphysical numerical solutions that may appear under certain circumstances, such as the entropy and the wet-dry front problems.…”