Well‐balancing issues related to the RKDG2 scheme for the shallow water equations

Abstract

SUMMARYDiscontinuous Galerkin (DG) finite element methods have salient features that are mainly highlighted by their locality, their easiness in balancing the flux and source term gradients and their component-wise structure. In the light of this, this paper aims to provide insights into the well-balancing property of a second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method. For this purpose, a Godunov-type RKDG2 method is presented for solving the shallow water equations. The scheme is based on local DG linear approximations and does not entail any special treatment of the source terms in order to achieve well-balanced numerical results. The performance of the present RKDG2 scheme in reproducing conserved solutions for both free surface and discharge over strongly irregular topography is demonstrated by applying to several hydraulic benchmarks. Meanwhile, the effects of different slope limiting procedures on the well-balancing property are investigated and discussed. This work may provide useful guidelines for developing a well-balanced RKDG2 numerical scheme for shallow water flow simulation.

Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

“…We directly outline a well-balanced RKDG2 scheme for solving the SWE [28] and then focus on the issues related to practical hydrodynamic modelling, i.e. …”

“…We consider a quiescent flow test case in a 1500m long frictionless domain with an irregular topographic profile that contains non-differentiable points [28]. The domain is first assumed Fig.…”

Topography discretization techniques for Godunov-type shallow water numerical models: a comparative study

“…The Hydr-Rec method reconstructs the free-surface elevation -likewise to the SGM -but acts differently to balance the flux and topography gradients so that to further maintain the positivity of the water depth, and as such provides the further ability to cope with wetting and drying. The Hydr-Rec approach has been successfully applied-as a topography discretization technique -to various high-order Godunov-type models such as WENO-FV and DG methods (Xing andShu 2006, Noelle et al 2007), and further enhanced -as a wetting and drying condition -with both FV and DG second-order Godunov-type models (Liang 2010, Kesserwani andLiang 2012). As to the DG method, Xing and Shu (2006) demonstrated theoretically that it is by far the simplest approach to obtain the well-balanced shallow water numerical model.…”

“…As to the DG method, Xing and Shu (2006) demonstrated theoretically that it is by far the simplest approach to obtain the well-balanced shallow water numerical model. In light of this, Kesserwani et al (2010) devised a second-order Runge-Kutta DG method (RKDG2) that is effortlessly well-balanced in which the local discrete topography is taken as piecewise-linear but globally continuous. This topography setting is particular to the RKDG2 framework which is experienced to be more costly than FV-based Godunov-type model.…”

Differential Quadrature Method for Linear Long Wave Propagation in Open Channels

“…Additionally, Zhou et al (2007) are proposed a split-characteristic Finite Element Model for 1-D unsteady flows. In the solution of this method, such as RDKG2 scheme (Kesserwani et al;2009), the modified Godunov method (Savic & Holly, 1993) and the Petrov-Galerking finite element scheme (Yang et al 1993;Khan 2000) has been used. In recently, use of Finite Volume Method is increasing.…”