A practical implementation of high‐order RKDG models for the 1D open‐channel flow equations

Abstract

SUMMARYThis paper comprises an implementation of a fourth-order Runge-Kutta discontinuous Galerkin (RKDG4) scheme for computing the open-channel flow equations. The main features of the proposed methodology are simplicity and easiness in the implementation, which may be of possible interest to water resources numerical modellers. A version of the RKDG4 is blended with the Roe Riemann solver, an adaptive high-order slope limiting procedure, and high-order source terms approximations. A comparison of the performance of the proposed method with lower-order RKDG models is performed showing a benefit of considering the RKDG4 model. The scheme is applied to computerize the Saint Venant system by considering benchmark tests that have exact solutions. Finally, numerical results are illustrated discussing the performance of the proposed high-order model.

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

“…The discharge solutions, however, have some oscillations as also observed in other methods [14,41,13,31,20,21] with a comparable magnitude. We also observe that the numerical oscillations in the discharge of the steady flows presented here are slightly larger than those in [41].…”

“…This type of tests have been used to examine if the numerical solution can converge to the steady state under the effect of bottom topography in literature [14,41,13,31,20,21] Depending on the boundary conditions at the two ends of the domain, different regimes of the final steady state can be obtained. Same as in [32,41], we consider the following three cases by imposing different boundary conditions.…”

Conservative finite-volume forms of the Saint-Venant equations for hydrology and urban drainage

“…In contrast, hydraulic studies have typically focused on 1 to 10 m spacing for 1 to 2 km test cases (e.g. Gottardi and Venutelli, 2003;Kesserwani et al, 2009;Sart et al, 2010;Venutelli, 2006). Between these extremes, single-reach river models with natural geometry are typically modeled over river lengths less than 20 km with grid cells on the order of 10 m to more than 100 m (Sanders et al, 2003;Catella et al, 2008;Castellarin et al, 2009;Lai and Khan, 2012).…”

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

“…A DG spatial framework starts with the local averages of flow information together with their corresponding local (natural) high-order slopes on a three-cell scheme. Even for simulations involving locally steep gradients of the flow variables that require a slope limiting process, computation is at most performed on a five-cell stencil [8]. DG methods, therefore, provide an attractive alternative and probably a more efficient way for solving the shallow water equations or the so-called Saint Venant equations.…”

“…Crossely and Wright [23] combined the local time stepping techniques and the upwind treatment of source terms to achieve well-balancing. Yet, source term upwinding is not straightforward for the high-order RKDG methods due to the necessity of dealing with high-order approximations of the source term integrals [8].…”