In recent years, discontinuous Galerkin (DG) methods have received ever increasing interest in the context of computational fluid dynamics, both for incompressible and compressible flows. Here, we will focus on unsteady compressibble flows. DG methods can be seen as generalizations of first order finite volume schemes, that extend to higher orders in a natural way. However, they have more severe stability restrictions for explicit time integration and thus, implicit schemes are necessary for a large set of applications. Of interest for unsteady flows are higher order schemes and we will apply diagonally implicit Runge-Kutta (DIRK) methods. This means that on each stage, a nonlinear equation system has to be solved. However, so far efficient solution schemes in the context of DG methods are lacking and the design of such algorithms is currently one of the most important challenges for the successful application of DG schemes.
In this talk, we will consider preconditioned Newton-Krylov schemes for
the solution of the resulting linear equation systems. The system
matrices are nonnormal block matrices, with block sizes of 16-120 for two
dimensional flows, which makes special treatment of these blocks
necessary and storing the whole matrix impossible for large problems.
Therefore, we consider matrix-free methods that circumvent storing the
Jacobian by approximating 
via difference quotients. Furthermore,
this allows to approximate the impact of the Jacobian by a cheap low
order operator in a straight forward way. Several preconditioners
suitable for matrix-free schemes will be considered, in particular Jacobi
and SGS.