We present a new parallel algorithm for the solution of the incompressible two- and three-dimensional Navier-Stokes equations. The parallelization is achieved via domain decomposition. The computational region is considered in the form of a 2-D or 3-D periodic box decomposed into parallel strips (slabs). For time discretization we use a third order multistep method. The time discretization procedure results in solving global elliptic problems of (monotonic) Helmholtz and Poisson types in each time step. For the space discretization we employ the multidomain local Fourier (MDLF) method. The discretization in the periodic directions is performed by the standard Fourier method. In the direction across the strips we use the Local Fourier Basis technique which involves the overlapping of the neighboring subdomains and smoothing of local functions across the interior boundaries (interfaces). The matching of the local solutions is performed by adding properly weighted interface Green's functions. Their amplitudes are found in terms of the jumps of the solution and its first derivatives at the interfaces.

Without the pressure term in each time step only the Helmholtz type equations were solved. It was shown that the parallel solution of this equation can be accomplished using only local (neighbor-to-neighbor) communication due to localization properies of the Helmholtz operator.

We here consider the complete Navier-Stokes system including the pressure term. The solution of the Poisson equation for pressure has the potential to degrade the performance and the achieved speedup of a parallel algorithm due to the global nature of this equation that necessitates global communication among the processors. However, we show that only a few lowest harmonics require the global data trasfer whereas the rest of harmonics can be treated locally. Therefore, most of the communication that is required for parallelization of the Navier-Stokes solver using the MDLF method is mainly local between adjacent subdomains (processors). Moreover, the percentage of the time spent in global communication reduces as the size of the problem increases. Thus, the present parallel algorithm is highly scalable.

The 2-D and 3-D Navier-Stokes solvers are implemented on three MIMD message-passing multiprocessors (a 60-processors IBM SP2, a 20-processors MOSIX, and a network of 10 Alpha workstations) and achieve an efficiency of more than 70% to 95%. The same code written with the PVM (parallel virtual machine) software package was executed on all the above distinct computational platforms.

This is joint work with L. Vozovoi, M. Israeli and L. Ioffe.

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1996-1997 Mathematics in High Performance Computing