Section author: Andreas Dedner <a.s.dedner@warwick.ac.uk>, Robert Klöfkorn <robert dot klofkorn at math dot lu dot se>, Martin Nolte <nolte.mrtn@gmail.com>

# Discontinuous Galerkin Methods: DUNE-FEM-DG Module¶

The add on module Dune-Fem-Dg provides a number of DG algorithms focusing on solving systems of evolution equations of advection-diffusion-reaction type of the form

$\partial_t U + \nabla\cdot\Big( F_c(x,t,U) - F_v(x,t,U,\nabla U) \Big) = S(U).$

The implemented method include a wide range of methods for DG discretization of the diffusion term including CDG2, BR2, IP, and many others. The advection term can be discretized using a local Lax-Friedrichs flux, specialized fluxes e.g. HLLE for the Euler equations, or user defined fluxes. To stabilize the DG method for advection dominated problems limiters with troubled cell indicators are available. Finally we use a method of lines approach for the time stepping based on explicit, implicit or, IMEX Runge-Kutta schemes using matrix-free Newton-Krylov solvers. As a final note the module can also be used to solve first order hyperbolic problems using a Finite-Volume method with linear reconstruction. A detailed description of the module is found in the [DK21].

Todo

dune-fem-dg examples need more details in notebooks