Riemann Solvers with Runge–Kutta Discontinuous Galerkin Schemes for the 1D Shallow Water Equations

The spectrum of this survey turns on the evaluation of some eminent Riemann solvers (or the so-called solver), for the shallow water equations, when employed with high-order Runge–Kutta discontinuous Galerkin (RKDG) methods. Based on the assumption that: The higher is the accuracy order of a numerical method, the less crucial is the choice of Riemann solver; actual literature rather use the Lax-Friedrich solver as it is easy and less costly, whereas many others could be also applied such as the Godunov, Roe, Osher, HLL, HLLC, and HLLE. In practical applications, the flow can be dominated by geometry, and friction effects have to be taken into consideration. With the intention of obtaining a suitable choice of the Riemann solver function for high-order RKDG methods, a one-dimensional numerical investigation was performed. Three traditional hydraulic problems were computed by this collection of solvers cooperated with high-order RKDG methods. A comparison of the performance of the solvers was carried out discussing the issue of L1-errors magnitude, CPU time cost, discontinuity resolution and source term effects.