Numerical Solutions of Euler Equations by Runge-kutta Discontinuous Galerkin Method

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 LaxFriedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for the Euler equations of gas dynamics from the viewpoint of kinetic theory. The aim is also to define a way of taking into account high-order space discretization in limiting process, to make use of all the expansion terms of the approximate solution. A limiting procedure is carefully designed to suppress numerical oscillations. It is demonstrated by the numerical experiments that the proposed RKDG methods give higher resolution in solving problems with smooth solutions. Moreover, shock and contact discontinuities can be well captured by using the proposed methods.