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

Abstract

In this paper 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 of [11]. 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 that was developed in [9, 10, 13]. 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.

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 not include the pressure term. Correspondingly, in each time step only the Helmholtx 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 properties of the Helmholtz operator. We 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 for the global data transfer 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.

2. In [l] we considered only 2-D equations. In this paper we extend the previous technique to 3-D problems.

3. Previously, the MDLF solver was implemented only on the MEIKO parallel machine. In this paper 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 [3], 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 [7]) software package was executed on all the above distinct computational platforms. Detailed performance results, which include scalability analysis, are presented.