Time integration in the discontinuous galerkin code MIGALE - unsteady problems

This chapter presents recent developments of a high-order Discontinuous Galerkin (DG) method to deal with unsteady simulation of turbulent flows by using high-order implicit time integration schemes. The approaches considered during the IDIHOM project were the Implicit Large Eddy Simulation (ILES), where no explicit subgrid-scale (SGS) model is included and the DG discretization itself acts like a SGS model, and two hybrid approaches between Reynolds-averaged Navier- Stokes (RANS) and Large Eddy Simulation (LES) models, namely the Spalart-Allmaras Detached Eddy Simulation (SA-DES) and the eXtra- Large Eddy Simulation (X-LES). Accurate time integration is based on high-order linearly implicit Rosenbrock-type Runge-Kutta schemes, implemented in the DG code MIGALE up to sixth-order accuracy. Several high-order DG results of both incompressible and compressible 3D turbulent test cases proposed within the IDIHOM project demonstrate the capability of the method.