Highly Scalable Two- and Three-Dimensional Navier-Stokes Parallel Solvers on MIMD Multiprocessors

• Authors: A. Averbuch;L. Ioffe;M. Israeli;L. Vozovoi
• Venue: The Journal of Supercomputing
• Year: 1997

In this paper we present a new parallel algorithm for the solution of theincompressible two- and three-dimensional Navier-Stokes equations. Theparallelization is achieved via domain decomposition. The computationalregion is considered in the form of a 2-D or 3-D periodic box decomposedinto parallel strips (slabs). For time discretization we use a third ordermultistep method of [11]. The time discretization procedure results insolving global elliptic problems of (monotonic) Helmholtz and Poisson typesin each time step. For the space discretization we employ the multidomainlocal Fourier (MDLF) method that was developed in [9, 10, 13]. Thediscretization in the periodic directions is performed by the standardFourier method. In the direction across the strips we use the Local FourierBasis technique which involves the overlapping of the neighboring subdomainsand smoothing of local functions across the interior boundaries(interfaces). The matching of the local solutions is performed by addingproperly weighted interface Green‘s functions. Their amplitudes are found interms of the jumps of the solution and its first derivatives at theinterfaces.The present paper extends the results of our previous works [1, 9, 10,13] on parallel use of the MDLF method in three-fold aspects:1. In [1] a model Navier-Stokes type system was considered which does notinclude the pressure term. Correspondingly, in each time step only theHelmholtx type equations were solved. It was shown that the parallelsolution of this equation can be accomplished using only local(neighbor-to-neighbor) communication due to localization properties of theHelmholtz operator. We consider the complete Navier-Stokes system includingthe pressure term. The solution of the Poisson equation for pressure has thepotential to degrade the performance and the achieved speedup of a parallelalgorithm due to the global nature of this equation that necessitates globalcommunication among the processors. However, we show that only a few lowestharmonics require for the global data transfer whereas the rest of harmonicscan be treated locally. Therefore, most of the communication that isrequired for parallelization of the Navier-Stokes solver using the MDLFmethod is mainly local between adjacent subdomains (processors). Moreover,the percentage of the time spent in global communication reduces as the sizeof the problem increases. Thus, the present parallel algorithm is highlyscalable.2. In [l] we considered only 2-D equations. In this paper weextend the previous technique to 3-D problems.3. Previously, the MDLF solver was implemented only on the MEIKO parallelmachine. In this paper the 2-D and 3-D Navier-Stokes solvers are implementedon three MIMD message-passing multiprocessors (a 60-processors IBM SP2, a20-processors MOSIX [3], and a network of 10 Alpha workstations) and achievean efficiency of more than 70% to 95%. The same code writtenwith the PVM (parallel virtual machine [7]) software package was executed onall the above distinct computational platforms. Detailed performanceresults, which include scalability analysis, are presented.