A numerical study for the performance of the Runge–Kutta discontinuous Galerkin method based on different numerical fluxes

Runge–Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge–Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax–Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist–Osher flux, etc., and second-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.