DEPARTMENT OF COMPUTER SCIENCE

DOCTORAL DISSERTATION DEFENSE

Candidate: Hadil G. Sabbagh

Advisor: Charles S. Peskin

DOCTORAL DISSERTATION DEFENSE

Candidate: Hadil G. Sabbagh

Advisor: Charles S. Peskin

**Solving the Navier-Stokes Equations on a Distributed
Parallel Computer**

11:00 a.m., Tuesday, April 23, 1996

room 402, Warren Weaver Hall

Abstract

Speed and space are two major issues in computational fluid dynamics today. Scalable parallel or distributed computers offer the promise of faster time to solve problems through parallelism and solutions to larger problems by adding more parallel processors with their own private memories. These systems use message passing to share data between processors. Parallel programs are difficult to write, especially for message passing systems, and there are few well-studied test cases.

In this dissertation, we solve the incompressible Navier-Stokes equations on a periodic cubic domain (3-torus). The numerical method is a finite difference method that consists of two parts: upwind differencing applied to the non-linear terms and solution of the Stokes equations. The latter are solved implicitly using a three-dimensional FFT. For the parallel implementation, the domain is divided into equally sized non-periodic cubic subdomains. Each subdomain is assigned to a processor; the processors form a process grid which is periodic. The parallel upwind differencing is preceded by an exchange of face data. The discrete Fourier transform in the Stokes solver is computed by applying one-dimensional FFTs sequentially in the three coordinate directions. In each coordinate direction, data must be exchanged only among those processors that lie on the same line of the process grid.

The parallel algorithm was implemented twice: once using PVM and once using MPI. Although both implementations are described in the thesis, the performance of only the MPI version is discussed.

The Navier-Stokes solver is tested on the IBM SP-2.
Three constant problem size and three scalability experiments are used to
analyze the performance of the solver.
The fluid solver achieves a speedup of 48.8 when solving a
240 * 240
* 240 problem on 216 processors.
Furthermore, there is evidence of scalability.