Application of a second‐order Runge–Kutta discontinuous Galerkin scheme for the shallow water equations with source terms

Abstract

AbstractThe present work addresses the numerical prediction of discontinuous shallow water flows by the application of a second‐order Runge–Kutta discontinuous Galerkin scheme (RKDG2). The unsteady flow of water in a one‐dimensional approach is described by the Saint Venant's model which incorporates source terms in practical applications. Therefore, the RKDG2 scheme is reformulated with a simple way to integrate source terms. Further, an adequate boundary conditions handling, by the theory of characteristics, was overviewed to be adapted to the external points of the mesh, as well as to some points of local invalidity of the Saint Venant's model. To validate the proposed technique, steady and transient test problems (all having a reference solution) were considered and computed by means of the overall method. The results were illustrated jointly with the reference solution and the results carried out by a traditional second‐order finite volume (FV2) scheme implemented with the same techniques as the RKDG2. The proposed method has proven its practical consideration when solving discontinuous shallow water flow involving: non‐prismatic channels, various cross‐sections, smoothly varying bed topography and internal boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.