The Navier-Stokes equations are discretized on staggered grids using the finite volume method, and are solved with the pressure implicit splitting-of-operators (PISO) algorithm which is an implicit time marching predictor-corrector algorithm based upon the splitting of the solution of velocity and pressure equations. One predictor and two corrector steps are sufficient to obtain a suitable accuracy. Then we have seven large sparse linear systems to solve at each time step. The evolutive transport and the potential equations are discretized by appropriate finite difference methods. In the work presented, the transport equation is solved using an implicit and an explicit time marching scheme. When an implicit scheme is used for the previous equation, we obtain two additional large sparse linear systems to solve. From the mathematical point of view, the matrices derived from discretizations are M-matrices; according to theoretical results, the convergence of the parallel synchronous and asynchronous Schwarz alternating method is ensured [1,2].

The experimental results of the numerical simulations show comparisons between asynchronous and synchronous methods. Implementation has been performed in Fortran 90 using MPI facilities and is based on the master-slave paradigm. In order to simplify the implementation, the master process builds matrices, right hand sides and activates a parallel linear system solver.

The experimental platform is the French grid platform GRID5000. We have to solve seven large linear systems at each step time. Experiments show that high acceleration is obtained from the parallel implementations. Further comparisons between synchronous and asynchronous algorithms show that the latter performs better than the former one. The use of an asynchronous parallel solver in a distributed cluster environment is a good approach since it involves no idle communication time.

1

M. Chau, P. Spiteri, R. Guivarch, H.C. Boisson, "Parallel asynchronous iterations for the solution of a 3D continuous flow electrophoresis problem", Computers and Fluids, 37(9), 1126-1137, 2008. doi:10.1016/j.compfluid.2007.06.006

2

D. El Baz, A. Frommer, P. Spitéri, "Asynchronous iterations with flexible communication: contracting operators", Journal of Computational and Applied Mathematics, 176, 91-103, 2005. doi:10.1016/j.cam.2004.07.009

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