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POISSON SOLVERS ON A LARGE ARRAY COIMPUTER

IP.com Disclosure Number: IPCOM000128256D
Original Publication Date: 1978-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 23 page(s) / 65K

Publishing Venue

Software Patent Institute

Related People

Chester E. Grosch: AUTHOR [+3]

Abstract

A recent analysis of the performance of a proposed large array computer for solving numerically the Navier-Stokes equations showed that the major portion of the time was spent solving the Poisson equation for the pressure field.1 it was assumed, in the analysis, that a parallel relaxation scheme (Red-Black SOR) was used to solve the Poisson equation at each time step. It is well known that, on serial computers, direct methods are more efficient than relaxation methods for the solution of the Poisson equation. This is not necessarily true for the proposed array computer because of the archi- tecture. The memory is distributed throughout the array and data transfers are only possible between nearest neighbor elements in the array; hence long range data transfer is very time consuming. The performance of three classes of Poisson solvers, standard parallel relaxation methods, parallel multi-grid relaxation methods, and parallel direct methods, on this array computer are analyzed. The analysis includes both the arithmetic operat.ion time and the data transfer time imposed by the array archi , tecture.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

POISSON SOLVERS ON A LARGE ARRAY COIMPUTER

by

Chester E. Grosch Institute of Oceanography and Department of Mathematical and Computing Sciences Old Dominion University Norfolk* Virginia

ABSTRACT

A recent analysis of the performance of a proposed large array computer for solving numerically the Navier-Stokes equations showed that the major portion of the time was spent solving the Poisson equation for the pressure field.1 it was assumed, in the analysis, that a parallel relaxation scheme (Red-Black SOR) was used to solve the Poisson equation at each time step. It is well known that, on serial computers, direct methods are more efficient than relaxation methods for the solution of the Poisson equation. This is not necessarily true for the proposed array computer because of the archi- tecture. The memory is distributed throughout the array and data transfers are only possible between nearest neighbor elements in the array; hence long range data transfer is very time consuming. The performance of three classes of Poisson solvers, standard parallel relaxation methods, parallel multi-grid relaxation methods, and parallel direct methods, on this array computer are analyzed. The analysis includes both the arithmetic operat.ion time and the data transfer time imposed by the array archi , tecture.

1. INTRODUCTION

The numerical calculation of turbulent flow fields is one of the major research areas in fluid dynamics. These calculations can be divided, quite roughly, into two classes: true simulations which.resolve all of the dynam-ically significant scales of motion; and phenomenological calculations which make use of turbulence models. The computing power of existing computers is the effective limit on both classes of calculations. There is a need, for both these types of calculation, to increase the number of mesh points (or miodes, for spectral calculations) in the flow volume and to reduce the average

........... computation time per mesh point. This requires an increase in both the memory size and
speed of computerg.

Increasing the memory size Allows the use of a finer spatial resolution for a fixed volume, the calculation of a flow in a larger volume at fixed resolution, the use of a more complex, and presumably more accurate, turbu--lence model, or some combination of these improvements. It is, of course, well known that the number of operations increases at a slightly faster rate than the number of mesh points, so that large problems require substantially more computational effort than small problems.

There have been some major improvements in algorithms; perhaps the most important are the development of efficient spectral methods, via the Fast Fourier Transform, 3 and the various algorithms for the efficient direct solu-tion of the Poisson Problem. 4-6 Quite recently a n I ew

Old Dominion University Page 1 Dec 31, 1978

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POISSON SOLVERS ON A LARGE ARRAY COIMP...