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

Abstract

SUMMARYThe 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.

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.…”

One‐dimensional simulation of supercritical flow at a confluence by means of a nonlinear junction model applied with the RKDG2 method

“…The proposed discretization is reported in detail in Reference [8]. We seek a local approximation (piecewise linear) U h to U that belongs to the finite dimensional space P k (I i ) of polynomial in I i of degree at most k = 1 leading to second-order accuracy in space.…”

“…Therefore, system (1) is multiplied by an arbitrary smooth function and integrated over I i . Then, the flux term is integrated by part to obtain the weak formulation (see [8,10,40]). With the aim of decoupling the system, we adopt the Legendre polynomials as a local basis function over I i .…”

“…However, applying explicit methods to simulate flows through a system of compound channels could prove advantageous by using a minimum of computational cells that leads to a reasonable simulation time cost without generating diffusions in the numerical solutions. Therefore, supported by relevant references reported in literature [8,11,33], the FE-type RKDG2 scheme is chosen for the approximation of the numerical solutions in channels. Furthermore, a supercritical benchmark having an analytical solution is selected [34] for the purpose of comparing the effectiveness of the RKDG2 model with two TVD-FV methods that were implemented with the same properties as RKDG2.…”