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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Peter Minev, University of Alberta
A Direction Splitting Algorithm for Flow Problems in Complex/Moving Geometries


An extension of the direction splitting method for the incompressible Navier-Stokes equations proposed in [1] to flow problems in complex, possibly time dependent geometries will be presented. The idea stems from the fictitious domain/penalty methods for flows in complex geometry. In our case, the velocity boundary conditions on the domain boundary are approximated 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 unconditionally stable for the Stokes problem and retains the same convergence rate in both, time and space, as the Crank-Nicolson scheme. A key advantage of this approach is that the algorithm has a very impressive parallel performance since it requires the solution of one-dimensional problems only, which can be performed very efficiently in parallel by a domain decomposition Schur complement approach. Numerical results illustrating 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.


  1. J.L. Guermond, P.D. Minev, A new class of massively parallel direction splitting for the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 200 (2011), 2083-2093.