PIMS-CSC Seminar: Peter Minev

With the advance of huge distributed parallel clusters many
of the current discretizations of parabolic problems are becoming slow and inefficient which opens the need for the development of new techniques. The usual bottleneck of the parallel algorithms for
unsteady incompressible problems is the solution of the pressure
Poisson equation which is the classical paradigm for enforcing incompressibility, proposed by the so-called projection methods. Our answer to this challenge was to resort to the very old technique of direction splitting, however, adapting it to the incompressibility constraint as well. Using a particular perturbation of the continuity equation we developed a technique for incompressible flow that requires the solution of one dimensional problems only. These  problems can be solved with a tridiagonal direct solver on a massive parallel cluster with a Schur complement technique. The resulting technique is unconditionally stable in case of the generalized Stokes problem and stable under the usual CFL constraint for transport problems, in case of the Navier-Stokes equations. Its computational expenses on a large parallel machine are fully comparable to the expenses of a fully explicit technique.

This talk will first focus on the convergence properties and the
implementation of such methods in the case of simple rectangular or
parallelepiped geometry. Then it will discuss the extension of this
technique to the case of flow problems in complex, possibly time
dependent geometries. The idea of this extension stems from the
fictitious domain/penalty methods for flows in complex geometries. In
our case, the velocity boundary conditions on the domain boundary are pproximated with a second-order of accuracy while the pressure
subproblem is harmonically extended in a fictitious domain such that
the overall domain of the problem is of a simple rectangular/ parallelepiped shape. The new technique is still nconditionally stable for the Stokes problem and retains the same onvergence rate as the Crank- Nicolson scheme. Numerical results llustrating the  convergence of the scheme in space and time will be presented. Finally, the implementation of the scheme for particulate flows will be discussed and some validation results for such flows will be presented.