Table Of ContentSpringer Series in Computational Physics
Editors
w. Beiglbock H. Cabannes
H. B. Keller 1. Killeen S. A. Orszag
F ini te-Difference
Techniques for Vectorized
Fluid Dynamics Calculations
Edited by David L. Book
With Contributions by
Jay P. Boris
Martin J. Fritts
Rangaro V. Madala
B. Edward McDonald
Niels K. Winsor
Steven T. Zalesak
With 60 Illustrations
Springer-Verlag
New Yark Heidelberg Berlin
David L. Book
Naval Research Laboratory
Washington, DC 20375
USA
Editors
w. Beiglbock H. Cabannes
Institut fiir Angewandte Mathematik Mecanique Theoretique
Universitat Heidelberg Universite Pierre et Marie Curie
D-6900 Heidelberg I F-75005 Paris
Federal Republic of Germany France
H. B. Keller J. Killeen
California Institute of Technology Lawrence Livermore Laboratory
Pasadena, CA 91125 Livermore, CA 94551
USA USA
S. A. Orszag
Massachusetts Institute of Technology
Cambridge, MA 02139
USA
Library of Congress Cataloging in Publication Data
Main entry under title:
Finite-difference techniques for vectorized fluid
dynamics calculations.
(Springer series in computational physics)
Bibliography: p.
Includes index.
I. Fluid dynamics. 2. Finite differences.
I. Book, D. L. (David Lincoln), 1939-
II. Series.
TA357.F49 532'.05'0151562 81-5282
AACR2
All rights reserved. No part of this book may be translated or reproduced in any
form without written permission form Springer-Verlag, 175 Fifth Avenue, New York,
New York 10010, U.SA.
© 1981 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1981
ISBN-13: 978-3-642-86717-0 e-ISBN-13: 978-3-642-86715-6
DOl: 10.1007/978-3-642-86715-6
Summary
This book describes several finite-difference techniques developed recently
for the numerical solution of fluid equations. Both convective (hyperbolic)
equations and elliptic equations (of Poisson's type) are discussed. The em
phasis is on methods developed and in use at the Naval Research Laboratory,
although brief descriptions of competitive and kindred techniques are included
as background material. This book is intended for specialists in computational
fluid dynamics and related subjects. It includes examples, applications and
source listings of program modules in Fortran embodying the methods.
Contents
Introduction 1
(D. L. Book)
2 Computational Techniques for Solution of Convective Equations 5
(D. L. Book and J. P. Boris)
2.1 Importance of Convective Equations 5
2.2 Requirements for Convective Equation Algorithms 7
2.3 Quasiparticle Methods 10
2.4 Characteristic Methods 13
2.5 Finite-Difference Methods 15
2.6 Finite-Element Methods 20
2.7 Spectral Methods 23
3 Flux-Corrected Transport 29
(D. L. Book, J. P. Boris, and S. T. Zalesak)
3.1 Improvements in Eulerian Finite-Difference Algorithms 29
3.2 ETBFCT: A Fully Vectorized FCT Module 33
3.3 Multidimensional FCT 41
4 Efficient Time Integration Schemes for Atmosphere and Ocean
Models 56
(R. V. Madala)
4.1 Introduction 56
4.2 Time Integration Schemes for Barotropic Models 58
4.3 Time Integration Schemes for Baroclinic Models 63
4.4 Extension to Ocean Models 70
David L. Book, Jay P. Boris, and Martin J. Fritts are from the Laboratory for Computational
Physics, Naval Research Laboratory, Washington, D.C.
B. Edward McDonald, Rangaro V. Madala, Niels K. Winsor, and Steven T. Zalesak are from
the Plasma Physics Division, Naval Research Laboratory, Washington, D.C.
viii Contents
5 A One-Dimensional Lagrangian Code for Nearly Incompressible
Flow 75
(J. P. Boris)
5.1 Difficulties Encountered in Lagrangian Methods 75
5.2 Adaptive Gridding in a Lagrangian Calculation 78
5.3 The Algorithm and Structure of ADINC 82
5.4 Examples 91
6 Two-Dimensional Lagrangian Fluid Dynamics Using Triangular
Grids 98
(M. J. Fritts)
6.1 Grid Distortion in Two Dimensions 98
6.2 Use of Reconnection to Eliminate Grid Distortion 100
6.3 Numerical Algorithms 105
6.4 Examples 112
7 Solution of Elliptic Equations 117
(R. V. Madala and B. E. McDonald)
7.1 Survey of Standard Techniques 117
7.2 A New Direct Solver: The Stabilized Error Vector Propagation
Technique (SEVP) 123
7.3 Application of Chebychev Iteration to Non-Self-Adjoint Equations 135
8 Vectorization of Fluid Codes 152
(N. K. Winsor)
8.1 Speed in Hardware 153
8.2 Speed in Fortran 154
8.3 Problems with Causality 157
8.4 Examples 159
8.5 Summary of Parallelism Principles 162
Appendix A 164
Appendix B 171
Appendix C 177
Appendix D 191
Appendix E 200
References 209
Index 215
CHAPTER 1
Introduction
In this book we describe a series of vectorizable computational techniques for
solving hydrodynamics problems which have been developed at the Naval
Research Laboratory in the past few years. We also relate them to older tech
niques, discuss the applications for which they were developed, and describe
their implementation on a particular vector machine, the NRL Texas Instru
ments Advanced Scientific Computer (ASC).
A tremendous range of physical phenomena are described by fluid equa
tions. All four of the classical "elements"-earth, water, air, and fire
behave, at least on some scales, as fluids. For this to be true, the features they
share in common must be very simple, in spite of the variegation and superficial
complexity we are accustomed to dealing with. This is indeed the case, and the
present book deals chiefly with those central aspects.
The mathematical description of fluid motion makes use of partial differen
tial equations which propagate the fluid variables (density, velocity, pressure,
etc.) in time. Most systems of such equations closely resemble one another in
containing convective derivatives, expressing causality, conservation laws,
and other properties, and in emphasizing the connectedness of the physical
system. This latter property is often manifested in the appearance of a Poisson
equation, which relates the pressure or velocity potential at a given point to
the properties of the physical system as a whole. It is also displayed in the
smooth (rather than discontinuous) variation through most of the system
which we expect in fluids. However, many of the most important computa
tional methods are intended to describe discontinuous behavior as well, such
as that occurring in shocks, at combustion fronts, and on free surfaces and
boundaries.
Because we have emphasized the common features and generality of fluid
behavior, most of the discussion in this book is theoretical, and the applica
tions given are fairly simple. To apply the techniques to specific problems of
interest, one or more of the following must be added to the basic (Navier
Stokes) fluid equations: (i) more dimensions; (ii) more complicated geometry
or boundary conditions; (iii) additional source, sink, or transport terms; (iv)
more dependent variables; (v) an equation of state. For example, ideal mag
netohydrodynamics requires the introduction of the magnetic field strength
B and the Lorentz force; nonideal magneto hydrodynamics requires thermal
2 1 Introduction
conduction, resistivity, and viscosity terms. Ocean internal waves require a
buoyancy term and an eddy viscosity derived from a turbulence closure
model. Chemically reactive flow requires the introduction of chemical vari
ables, together with reaction rates, thermal conduction, and a turbulence
model. Although most of the physics in any given problem is comprised in
these added terms and conditions, computational practice shows that most
of the difficult numerics problems are associated with the universal fluid
features. If adequate care is devoted to developing accurate general numerical
techniques for treating these, the resulting code invariably turns out to be
robust, adaptable, and efficient. Hence the techniques described here encom
pass a substantial portion of the effort needed to construct even the most
elaborate numerical hydoodynamic model.
We discuss techniques for both "incompressible" and "compressible"
hydrodynamics, distinguished according to whether V· v = 0 or V· v ¥- o.
In general we will be treating systems in which the fundamental variables are
the mass density p, flow velocity v, and pressure p, and the equations are,
respectively, those of continuity or advection; momentum transfer (Euler's
equation); and an energy transport equation, equation of state, or the incom
pressibility condition (see Section 2.1).
It is assumed that the reader is familiar with the basic concepts of fluid
dynamics, as expounded, for example, in Landau and Lifshitz (1959). No
attempt will be made in this book to study in any detail the physics of fluids,
or to explore the physical significance of the specialized problems which have
been solved using the computational methods we present. Instead the em
phasis is on the methods themselves.
The reader is expected to be conversant in the standard numerical tech
niques used for machine computation, as given in, e.g., Hamming (1962), and
to have had experience in programming physics problems in Fortran or some
other high-level language. Moreover, he should have some acquaintance with
standard methods for computer solution of fluid problems, such as would
result from having been involved in research in which fluid computations were
carried out or from having read the descriptions given by, e.g., Richtmyer and
Morton (1967) or Roache (1975). This book is intended, in other words, for
the specialist in computational fluid dynamics or in one of the fields which
makes use of fluid computational techniques: plasma, space, solar or astro
physics, oceanology, meteorology, combustion, etc.
As used in this book, "computational techniques" relate to the solution of
equations derived from theoretical models, specifically models of hydro
dynamic phenomena, and are to be distinguished from the use of computers
in data reduction and analysis. Almost all of the techniques are aimed at
solving nonlinear problems, which are inherently intractable by classical
analytic techniques. Indeed, it is to their nonlinearity that fluid dynamic sys
tems owe both their richness and their relative inaccessibility to computational
treatment.
1 Introduction 3
Every theoretical model yields its predictions as solutions of the model
equations. These may be as general as the three-dimensional Navier-Stokes
equations, or as specific as an equation for the evolution of a particular type
of wave (e.g., solitons) or a steady-state turbulent spectrum. Broadly speaking,
there is a correlation between generality and the following difficulties:
1. Heavy machine requirements (storage and CP time);
2. need for highly sophisticated numerical methods;
3. limitations in precision and validity of long-term predictions;
while specificity is associated with
1. theoretical problems in deriving and justifying the model;
2. stringent requirements on plausibility and agreement of predictions with
experiment.
In addition to determining the concrete predictions inherent in a particular
set of assumptions, computer modeling can be used in an exploratory role,
through "computer experiments." Where the precise set of assumptions or
the values of the physical parameters are not clear, it is possible to generate a
family of solutions to study trends and systematic properties. This form of
computation is often referred to as "numerical simulation" and is contrasted
with "numerical solution" of equations.
This computer-experimentation point of view is the one adopted in the
present work. We will be concerned principally with addressing the three
kinds of difficulty listed above in connection with general models. The
techniques we describe are designed to work in a production mode which
emphasizes convenient data structures, high resolution (large computational
meshes), detailed diagnostics, ruggedness, and, so far as is compatible with
these, high execution rates.
Most techniques which are inherently efficient show to good advantage on
any machine and can be coded in nearly machine-independent form. In order
to shave the last few percent (sometimes more than a few) from execution
times, it is essential to make optimum use of the characteristics ofthe individual
machine. The methods described in the text and the corresponding modules
listed in the Appendices were developed for the NRL ASC. While some have
been run on other machines with only small changes, optimization would
entail a certain amount of recoding. The effort required varies, depending on
both the machine and the routine involved.
This book is organized as follows. In Chapter 2 six different classes of
methods are compared and categorized in a framework provided by a set of
six requirements which ought to be satisfied by an ideal continuity equation
algorithm. The strengths and weaknesses of the different approaches are
analyzed. Within this framework finite-difference techniques have particularly
attractive properties. Hence most of the book is devoted to describing tech
niques which correct, at least partially, the difficulties from which finite-
4 1 Introduction
difference techniques suffer. Chapter 3 is devoted to flux-corrected transport
(FCT) and related Eulerian solutions of the nonlinear continuity equation,
which preserve the positivity property of the real equations and yet have
relatively low numerical diffusion. Chapter 4 describes another kind of
Eulerian code, designed to solve evolution equations describing the upper
ocean and the atmosphere without the stringent time-step limitation imposed
by an explicit treatment of surface waves and fast-moving internal gravity
waves. Chapter 5 describes ADINC, a Lagrangian algorithm appropriate
for describing nearly incompressible flow. In Chapter 6 we consider the
Lagrangian techniques incorporated in the SPLISH code, which uses a recon
nectable grid of triangles to allow accurate numerical solution of complicated
flows with breaking waves and other severe nonlinearities. In Chapter 7 we
briefly survey elliptic equation solvers, then describe two new techniques-one
direct and one iterative-which have proven useful in fluid computations.
Chapter 8 describes the rationale behind vectorization and summarizes
techniques used to vectorize codes we have written incorporating the tech
niques described in this book. From several of these codes we have extracted
the core modules, Fortran listings of which are collected in the Appendices,
along with directions for utilization.
While this book is in no sense a treatise on the whole subject of computa
tional fluid dynamics, it spans a wide range of topics. These were selected
because each represents a definite advance in the state of the computational
art. Although they have been described in specialist journals, each of the
techniques is useful in a wide variety of applications. By collecting them in this
fashion, relating them to each other and to previously existing techniques, and
describing both theory and applications, we hope to bring about their exposure
to a much wider audience than would otherwise be possible.
