Jean-Claude Latché has defended his HDR on 9th June 2010
at INSTN in Cadarache.
Prof. Pierre FABRIE (Univ. Bordeaux 1), referee
Prof. Jean-Luc GUERMOND (US-Texas A&M University), referee
Prof. Raphaële HERBIN (Univ. Aix-Marseille 1), member
Prof. Philippe ANGOT (Univ. Aix-Marseille 1), member
Prof. Thierry GALLOUET (Univ. Aix-Marseille 1), member
Prof. Peter D. MINEV (Univ. Alberta, Canada), member
In the first part of this thesis, we present a coherent family of pressure correction algorithms for the calculation of incompressible flow, and then at low Mach number, compressible flow, and finally, possibly two-phase flows. These schemes are based on low-degree spatial discretizations, where the degrees of freedom of the velocity are localized on the faces of the mesh and the other unknowns are constant per unit cell. The velocity-pressure pair is thus approximated by a Crouzeix-Raviart element for unit cells taking the form of a simplex, or a Rannacher-Turck for quadrilaterals (2D) or hexahedra (3D). Nevertheless, although the diffusion and pressure gradient terms in the conservation of momentum equation are discretized using a finite element technique, the others are approached via a finite volume method based on a dual mesh system. These schemes inherit stability properties from the continuous problem, and in particular they reveal an energy inequality irrespective of the time step. To the best of our knowledge, these stability results are the first of their kind for compressible flow. Moreover, we demonstrate the convergence of the Crouzeix-Raviart finite element approximation with a model problem, the compressible Stokes problem, which also appears to be the first numerical convergence result for these equations.
Secondly, we propose a new time algorithm for the incompressible or low Mach number case. This technique is called the ‘penalty-projection method’ and differs from the usual production schemes through the addition of a penalty term in the prediction step, which constrains the velocity to satisfy conservation of mass. This term is multiplied by a coefficient r, known as the penalty parameter. We use numerical experiments to show that this scheme is far more precise than the usual method. The splitting error, dominant when the time step is large, is reduced at will by increasing r. Note, however, that the use of too high a value degrades the conditioning of the operator associated with the prediction step. Moreover, the convergence losses of the usual projection method in the case of open boundary conditions are corrected, as long as r is nonzero. These digital experiments are supported by a theoretical study carried out for the unsteady Stokes flow problem, which shows that this scheme inherits time convergence properties from the two families of algorithm it combines: for low r, the splitting error behaves, in energy norms, like that of the recent variant of the projection method called the 'rotational method', i.e. like δt2 and δt3/2 for the velocity and pressure respectively; for high values of r, we obtain for the velocity error the classic δt/r increase for penalty methods, while the pressure error varies as 1/r.
Finally, for the incompressible Navier-Stokes equations, we construct schemes based on a different spatial discretization (and, for an equal number of unit cells, requiring fewer degrees of freedom): collocated finite volumes. The proposed schemes have the specific feature of stabilization terms added to the conservation of mass equation, i.e., in this case, to the zero-divergence constraint. We demonstrate its stability and convergence for the Stokes and Navier-Stokes equations, both steady and unsteady, and we prove optimum error increases for the steady Stokes problem. To the best of our knowledge, all of these results are new for this type of approximation in space.