High order discontinuous Galerkin methods for the Navier-Stokes equations

This thesis is devoted to the achievement of numerical methods for the solution of the Navier-Stokes equations, both compressible and incompressible, for fuid dynamics applications. The main objective has been to establish a fair trade-off between the accuracy requirement and the computational cost, since although the higher resolution methodologies can proven to supply advantages as regards the quality of the solution provided, in fact they are often claimed to be excessively costly. This work can be divided into two parts. First we present the implementation of a spectral discontinuous Galerkin methods for the compressible Navier-Stokes set. The co-location strategy in gaussian nodes along with the tensorial computation of the approximating functions has been implemented, for an effective assembly of the discrete operators. This approach has also been joined with a p-multigrid method achieving an overall quite interesting performance enhancement. An extensive numerical validation is provided in order to investigate both the accuracy and the e ciency of the scheme. The second part is dedicated to the solution of the incompressible Navier-Stokes equations. A new semi-implicit algorithm has been devised through a procedure of minimization of the implicitness level necessary to achieve the solvability of the problem. Such an aim required to additively split the system to outline and distinguish the implicitly considered parts from those evaluated explicitly. This scheme clearly takes advantage in term of the memory requirement.