# Parallel Computations

## 1st Edition

**Editors:**Garry Rodrigue

**eBook ISBN:**9781483276649

**Imprint:**Academic Press

**Published Date:**28th October 1982

**Page Count:**416

## Description

Parallel Computations focuses on parallel computation, with emphasis on algorithms used in a variety of numerical and physical applications and for many different types of parallel computers. Topics covered range from vectorization of fast Fourier transforms (FFTs) and of the incomplete Cholesky conjugate gradient (ICCG) algorithm on the Cray-1 to calculation of table lookups and piecewise functions. Single tridiagonal linear systems and vectorized computation of reactive flow are also discussed.

Comprised of 13 chapters, this volume begins by classifying parallel computers and describing techniques for performing matrix operations on them. The reader is then introduced to FFTs and the tridiagonal linear system as well as the ICCG method. Different versions of the conjugate gradient method for solving the time-dependent diffusion equation are considered. Subsequent chapters deal with two- and three-dimensional fluid flow calculations, paying particular attention to the principal issues in designing efficient numerical methods for hydrodynamic calculations; the decisions that a numerical modeler must make to optimize chemically reactive flow simulations; and how to handle disk-to-core data transfer and storage allocation for the solution of the implicit equations for three-dimensional flows. The book also describes the time-split finite difference scheme for solving the two-dimensional Navier-Stokes equation for flows through slotted nozzles. Finally, the large-scale stimulation of plasmas, as carried out on a small computer with an array processor, is discussed.

This monograph should be of interest to specialists in computer science.

## Table of Contents

List of Contributors

Preface

A Guide to Parallel Computation and Some Cray-1 Experiences

I. Introduction

II. Hardware

III. Theoretical Considerations

IV. Applications

Appendix A. A Register Assignment for Sparse-Banded Matrix Multiply

Appendix B. Factor and Forward Substitution

Appendix C. Backward Substitution

Appendix D. Factorization Only

References

Vectorizing the FFTs

I. Introduction

II. Preliminaries

III. The Complex FFT Algorithms

IV. Vectorizing Multiple Transforms

V. Transforming Real Sequences

VI. The Symmetric Transforms

VII. Software and Summary

References

Solution of Single Tridiagonal Linear Systems and Vectorization of the ICCG Algorithm on the Cray-1

I. A Vector Algorithm for Tridiagonal Linear Systems

II. An Incomplete Cholesky Conjugate Gradient (ICCG) Algorithm for the Cray-1 Computer

III. Cyclic Reduction on Future Machines

References

An Implicit Numerical Solution of the Two-Dimensional Diffusion Equation and Vectorization Experiments

I. Introduction

II. Spatial Differencing

III. Matrix Formulation

IV. Properties of the Matrix A

V. Method of Lines

VI. The Generalized Conjugate Gradient Algorithm

VII. Computational Example

VIII. Comments and Conclusions

References

Swimming Upstream: Calculating Table Lookups and Piecewise Functions

I. Introduction to Table Lookup

II. Evaluating Algorithms on Vector Processors

III. Basic Processes on Vector Processors

IV. One-Dimensional Problems

V. Two-Dimensional Problems: Equations of State

References

Trade-Offs in Designing Explicit Hydrodynamical Schemes for Vector Computers

I. Introduction

II. Why Vectorization of Explicit Hydrodynamical Schemes Should Be Easy

III. Why Vectorization of Explicit Hydrodynamical Schemes Can Be Difficult

IV. Alternative Approaches and Their Costs on Vector Computers 160

V. The Example of the Interaction of Two Blast Waves

VI. Conclusions

References

Vectorized Computation of Reactive Flow

I. Introduction and Statement of the Problem

II. Vectorization and Optimization

III. Techniques for Modeling Fast Time Scales

IV. Techniques for Modeling Short Space Scales

V. Techniques for Dealing with Physical and Geometric Complexity

VI. Programming Guidelines and Summary of Parallelism Principles

References

A Fully Implicit, Factored Code for Computing Three-Dimensional Flows on the ILLIAC IV

I. Introduction

II. Basic Equations

III. ILLIAC Architecture

IV. Data-Base Considerations

V. The ILLIAC Code ARC3

VI. Results

VII. Concluding Remarks

References

A Time-Split Difference Scheme for the Compressible Navier-Stokes Equations with Applications to Flows in Slotted Nozzles

I. Introduction

II. The Difference Scheme

III. The Application

IV. The Implementation

V. Results

Appendix. Numerical Grid Generation

References

Geophysical Fluid Simulation on a Parallel Computer

I. Introduction

II. The Salient Characteristics of the ASC

III. The FORTRAN Compiler on the ASC

IV. The Physical Processes of a Model

V. Estimating Parallelism in Models

VI. Conclusion

Experiences with a Floating Point Systems Array Processor

I. Introduction

II. Scientific Computing beyond the CDC 7600

III. The AP-190L Installation at Cornell

IV. FPS Array Processors and Parallel Computing

V. Examples of Optimal Programming for the AP

VI. The Two-Machine Environment

VII. Practical Problems of AP Ownership

VIII. Conclusions

References

A Case Study in the Application of a Tightly Coupled Multiprocessor to Scientific Computations

I. Introduction

II. Tightly Coupled Multiprocessors

III. Case Studies

IV. Conclusions

Appendix. Implementing Parallel Algorithms

References

Computer Modeling in Plasma Physics on the Parallel-Architecture CHI Computer

I. Introduction

II. Formulation of the Simulation Problems

III. Design of the Computer System

IV. Programming for Efficiency

V. Observations and Speculations

References

Index

## Details

- No. of pages:
- 416

- Language:
- English

- Copyright:
- © Academic Press 1982

- Published:
- 28th October 1982

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483276649