The Discontinuous Galerkin Method and its Implementation in the RAMEG3D Software Package

Abstract

Currently, the Discontinuous Galerkin Method (DGM) is widely used to solve complex multi-scale problems of mathematical physics that have important applied significance. When implementing it, the question of choosing a discrete approximation of flows for viscous terms of the Navier-Stokes equation is important.It is necessary to focus on the construction of limiting functions, on the selection of the best discrete approximations of diffusion flows, and on the use of implicit and iterative methods for solving the obtained differential-difference equations for the successful application of DGM on three-dimensional unstructured grids.First-order numerical schemes and second-order DGM schemes with Godunov, HLLC, Rusanov-Lax-Friedrichs numerical flows and hybrid flows are investigated. For high-order precision methods, it is necessary to use high-order time schemes.The Runge-Kutta scheme of the third order is used in the work. The equations are written as a system of first-order equations, when solving the Navier-Stokes equation by the discontinuous Galerkin method.