An agglomeration-based discontinuous Galerkin method for compressible flows

This thesis investigates the flexibility associated to Discontinuous Galerkin (DG) discretization on very general meshes obtained by means of agglomeration techniques. The work begins with a brief overview of the main tools that have been extended or specifically developed to deal with arbitrarily shaped elements in the DG context. Then two different implementations of the BRMPS scheme introduced by Bassi, Rebay, Mariotti, Pedinotti and Savini in [16] for the DG discretization of the Laplace operator on arbitrarily shaped elements have been presented. The validation of the scheme on a Poisson problem shows that the discrete polynomial space preserves optimal convergence properties. The discretization of the second order differential operator has been directly extended to the Navier-Stokes equations and the Reynolds Averaged Navier-Stokes (RANS) equations coupled with the k-w turbulence model of Wilcox [54]. In this regard, an implicit time integration strategy has been considered and assessed on classical validation test cases for the compressible f uid dynamics. Then a simple alternative approach to high-order mesh generation is presented. Indeed, once a standard fine grid able to provide an accurate domain discretization has been produced by means of standard low-order grid generation tools, a computational mesh suitable for the desired accuracy and computationally affordable can be obtained via agglomeration while keeping the boundary resolution of the fine grid. The effectiveness of this approach in representing the geometry of the domain is numerically assessed both on a Poisson model problem and on challenging inviscid and viscous test cases. Finally, the freedom in simply defining the topology of agglomerated meshes leads to a nonstandard approach to h-adaptivity that exploits adaptive agglomeration coarsening of a properly fine underlying grid. The effectiveness of this approach has been assessed on test cases involving both error-based and fl ow feature-based simple estimators.