Pacific Institute for the Mathematical Sciences - PIMS

Massively parallel discretizations for incompressible flow

With the advance of huge distributed parallel clusters manyof 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 forunsteady incompressible problems is the solution of the pressurePoisson 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 theimplementation of such methods in the case of simple rectangular orparallelepiped geometry. Then it will discuss the extension of thistechnique to the case of flow problems in complex, possibly timedependent geometries. The idea of this extension stems from thefictitious domain/penalty methods for flows in complex geometries. Inour case, the velocity boundary conditions on the domain boundary are pproximated with a second-order of accuracy while the pressuresubproblem is harmonically extended in a fictitious domain such thatthe 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.

Location: SFU- TASC-2, Rm 8500