The Use of Supercomputers in Stellar Dynamics: Proceedings of a Workshop Held at the Institute for Advanced Study Princeton, USA, June 2–4, 1986

233 Pages·1986·3.531 MB·English

by  Lyman Spitzer (auth.), Piet Hut, Stephen L. W. McMillan (eds.)

erutceL Notes
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in
Edited yb .H Araki, Kyoto, .J Ehlers, MSnchen, .K Hepp, ZSrich
.R Kippenhahn, MSnchen, H.A. Heidelberg WeidenmSIler,
.J Wess, Karlsruhe dna .J Zittartz, K~ln
Managing Editor: W. Beiglb6ck
267
I I II I II II IIII II II I II
The Use of Supercomputers
ni Stellar Dynamics
Proceedings of a Workshop
Held at the Institute for Advanced Study
Princeton, USA, June 2-4, 1986
Edited by .P Hut and S. McMillan
I III
galreV-regnirpS
Heidelberg Berlin NewYork London Paris oykoT
Editors
Piet Hut
Institute for Advanced Study
Princeton, NJ 08540, USA
Stephen L.W. McMillan
Drexel University
Philadelphia, PA 19104, USA
ISBN 3-540-17196-7 SpringeT-Verlag B~n Heidelberg NewYork
ISBN 0-387-17196-7 Springer-Versa 9 NewYork Berlin Heidelberg
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Preface
This introduction, as well as the following book, should not exist, according to our original
announcement. The meeting was advertised half a year ago as "an informal workshop on the
Use of Supercomputers in Stellar Dynamics, for which there will be no proceedings, no special
social or cultural events, and even no registration fee, only a registration form. What will there
be? Lots of informal discussions, a few brief and informal presentations with the main purpose
of triggering discussions on specific topics, and intermissions long enough to allow discussions
between individuals as well."
Soon after the announcement was made public, we received about a hundred applications,
which made us realize that we had to change our original plans. The good news was that we
had been both successful in choosing our topic and able to attract most researchers actively
participating within it. Accordingly, we decided to adapt our original scheme by relaxing one of
our three restrictions and voil& the result rests in your hands.
The meeting covered three days, each owfh ichh ad a distinct flavor, which can be summarized
sa Astrophysics, Architectures and Algorithms.
Astrophysics was the topic of the first day, in order to define the supercomputing problems in
their astrophysical context. Since this had more of a review character, only six invited speakers
were asked to give a contribution, while the rest of the time was spent according to plan: on
informal discussions. These six talks covered three major areas in stellar dynamics: the study
of (a) star clusters, (b) galaxies, (c) cosmology. Each of these areas have their own specific
kinds of astrophysical and computational problems, as well as their own types of techniques and
algorithms. These categories provided a natural choice of three morning talks about astrophysical
problems by (a) Spitzer, (b) Sellwood, and (c) Fall; and three afternoon talks by (a) Heggie, (b)
van Albada, and (c) Efstathiou.
Architectures, the topic of the d secor, day, was left largely to the invited representatives
from a number of companies, as well as academic groups involved in building new types of super-
or parallel computers. Included in the present volume are those contributions which reached us
before our final submission deadline. In the case of company representatives, the content oft hese
papers reflect only the views of the authors and their companies; no editorial advice on future
computer purchases is implied!
Although most of the architecture talks were given by non-astronomers, a notable exception
was the report by Gerald Sussman. He and co-workers from M.I.T. and Caltech have recently
constructed a special-purpose computer for the study of solar system dynamics. Since this effort is
unique, and is as far as we know the first such enterprise in the interface between astrophysics and
computer science, we have decided to include two reprints concerning his project in the present
proceedings: one on the design, and one on the first astrophysics results.
Algorithms were discussed on the third day, when individual researchers reported on their
hands-on experience as physicists using super/parallel computers. The tales of their troubles and
tribulations provided an interesting contrast to the often-heard glowing appraisal of supercomput-
ers in terms of Megafloppage, peak performance, and so on. Some of the long-term calculations
were actually performed on a small workstation left to run for a few months, with the drawback
of a large turn-around time, but the advantage of a minimal change in algorithm, data in/output,
etc. Other workers, however, reported how one can successfully put a supercomputer to good
use, once all the initial hurdles have been overcome. One aspect which was generally stressed was
VI
the hope and expectation that future computer facilities would not only increase in performance,
but also in ease of use, access and communication.
Participants in the workshop ranged from astrophysicists with little or no experience of
supercomputers to computer manufacturers with a similarly slight knowledge of astronomy. The
meeting was therefore a useful learning experience for all concerned. Many of the discussion
periods centered around the basic problem that "vanilla-flavored" computer codes can fail short
of their optimal running speed by an order of magnitude or more if care si not taken to implement
at least a modest amount of vectorization and parallelization. More so now than in the past, the
tailoring of algorithms to machines, as well as machines to algorithms, is becoming essential if
peak performance is to be attained. Judging from the number of "helpful" suggestions traded,
the time may be right for productive cooperation between computer designers and scientific users.
An interesting result emerging from the final discussion was the small number of qualitatively
new results that have so far come from supercomputers, notwithstanding their greater number-
crunching power. Instead, machines that are slower by one or two orders of magnitude have often
been used for proportionally longer periods of time to achieve the same ends. One reason for
this phenomenon is the widespread availability of minicomputers and workstations, which are
typically used by individuals or small groups of researchers, whereas supercomputers generally
are shared remotely by many users. Another, perhaps more important reason, is the additional
effort required to port one's code from a familiar operating system to a new (and traditionally
less than user-friendly) supercomputing em'ironment.
This latter difficulty will hopefully be overcome soon, with increasingly fast and convenient
high-speed communications and the adoption of a standard operating system (at present UNIX
seems to be the front runner). The prominence of high-speed communications and the support
of local workstations in the organization of the NSF supercomputer centers should be welcomed
by the scientific community. The former problem can only be addressed when supercomputer
time becomes more widely available, and when individual users with computer-intensive projects
can acquire the equivalent of a few VAX-years (i.e. a couple of hundred supercomputer hours)
without too much trouble. In this respect too, the NSF centers can fill an increasing need.
The scientific organizing committee for the workshop consisted of Sverre Aarseth, Joshua E.
Barnes, James J. Binney, Raymond G. Carlberg, Ortwin Gerhard, Douglas C. Heggie, Piet Hut
(chairman), Shogo Inagaki, Stephen L. W. McMillan, Peter J. Quinn, Gerald J. Sussman and
Scott D. Tremaine.
We acknowledge the enthusiastic and efficient help we have received from Michelle Sage,
without whose organizational skill and energy the workshop would not have been possible. We
also thank Mary Wisnowsky, the assistant to the director at the I.A.S., for her enthusiastic
support, and Sarah Johns for her help in the overall organization.
Piet Hut
Steve McMillan
TABLE OF CONTENTS
Session 1. ASTROPHYSICAL PROBLEMS AND MATHEMATICAL MODELS
L. Spitzer, Jr.:
Dynamical Evolution of Globular Clusters . . . . . . . . . . . . . . . . . 3
J.A. Sellwood:
Disc Galaxy Dynamics on the Computer . . . . . . . . . . . . . . . . . . 5
D.C. Heggie:
Star Cluster Dynamics: Mathematical Models . . . . . . . . . . . . . . . 13
T.S. van Albada:
Models of Hot Stellar Systems . . . . . . . . . . . . . . . . . . . . . . 23
G. Efstathiou:
Supercomputers and Large Cosmological N-Body Simulations . . . . . . . . 36
LELLARAP~REPUS
Session 2. COMPUTERS
R.A. James:
Modelling Stellar Dynamical Systems on the CRAY-1S and the CDC Cyber 205 49
C.N. Arnold:
Programming the ETA °1 for Large Problems in Stellar Dynamics . . . . . . . 54
J.L. Gustafson, S. Hawkinson and K. Scott:
The Architecture of a Homogeneous Vector Supercomputer . . . . . . . . . . 62
D.C. Allen:
The BBN Multiprocessors: Butterfly and Monarch . . . . . . . . . . . . . 72
D. Hillis:
The Connection Machine . . . . . . . . . . . . . . . . . . . . ..... 84
J.H. Applegate, M.R. Douglas, Y. Gfirsel, P. Hunter, C.L. Seitz and G.J. Sussman:
A Digital Orrery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
J.H. Applegate, M.R. Douglas, Y. Gfirsel, G.J. Sussman and J. Wisdom:
The Outer Solar System for 200 Million Years . . . . . . . . . . . . . . . 96
Session 3. CONTRIBUTIONS
W. Benz:
Smooth Particle Hydrodynamics: Theory and Application to the
Origin of the Moon . . . . . . . . . . . . . . . . . . . . . . . . . . 117
R.A. James and T. Weeks:
Multiple Mesh Techniques for Modelling Interacting Galaxies . . . . . . . . 125
P.J. Quinn, J.K. Salmon and W.H. Zurek:
Numerical Experiments on Galactic Halo Formation . . . . . . . . . . . . 130
IV
M. Lecar:
Numerical Integration Using Explicit Taylor Series . . . . . . . . . . . . 142
K.L. Chan, W.Y. Chau, C. Jessop and M. Jorgenson:
Multiple-Mesh-Particle Scheme for N-Body Simulation . . . . . . . . . . . 146
J. Makino:
Direct N-Body Simulation on Supercomputers . . . . . . . . . . . . . . 151
S.L.W. McMiIlan:
The Vectorization of Small-N Integrators . . . . . . . . . . . . . . . . . 156
M.J. Duncan:
N-Body Integrations Using Supercomputers . . . . . . . . . . . . . . . 162
R.L. White:
A New Numerical Technique for Calculation of Phase Space Evolution
of Stellar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
J.E. Barnes:
An Efficient N-Body Algorithm for a Fine-Grain ParalIel Computer ..... 175
G.B. Rybieki:
A Gridless Fourier Method . . . . . . . . . . . . . . . . . . . . . . . 181
W.H. Press:
Techniques and Tricks for N-Body Computation . . . . . . . . . . . . . 184
P. Hut and G.J. Sussman:
On Toolboxes and Telescopes . . . . . . . . . . . . . . . . . . . . . . 193
POSTER SESSION
S.J. Aarseth and E. Bettwieser:
A Unified N-Body Method 20t
. . . . . . . . . . . . . . . . . . . . . . .
S.J. Aarseth and S. Inagaki:
Vectorization of N-Body Codes . . . . . . . . . . . . . . . . . . . . . 203
H. Cohn, M.W. Wise, T.S. Yoon, T.S. Statler, J.P. Ostriker, and P. Hut:
Large Scale Calculations of Core Oscillations in Globular Clusters . . . . . . 206
H. Dejonghe and P. Hut:
Round-Off Sensitivity in the N-Body Problem . . . . . . . . . . . . . . 212
S.Y. Kim, H.M. Lee and K.W. Min:
Formation of a Bar Through Cold Collapse of a Stellar System . . . . . . . 219
M.C. Schroeder and N.F. Comins:
The Gravitational Interaction Between N-Body (Star Clusters) and
Hydrodynamic (ISM) Codes in Disk Galaxy Simulations . . . . . . . . . . 223
APPENDIX
D.C. Heggie and R.D. Mathieu:
Standardised Units and Time Scales . . . . . . . . . . . . . . . . . . . 233
LIST OF PARTICIPANTS . . . . . . . . . . . . . . . . . . . . . . . . 237
DYNAMICAL EVOLUTION OF GLOBULAR CLUSTERS
Lyman Spltzer, Jr.
Princeton University Observatory
Princeton, N.J. 08540
While research on the dynamical evolution of star clusters has been
underway for many years, substantial progress has been possible only during
the last two decades, since fast computers have been available. The advent of
still more powerful computers should much extend our understanding of this
field.
As an introduction to some of the problems for which supercomputers might
be applied, the present paper summarizes present knowledge of this field*. The
relevant physical processes and their effects on cluster evolution are
described and some of the principal questions for further research are listed.
The physical process chiefly responsible for dynamical evolution of
clusters is the tendency toward a Maxwellian distribution of random stellar
velocities. This tendency results from gravitational encounters between
pairs of stars, producing many small changes fo velocity and resultant
diffusion in velocity space. As a result of this tendency some stars tend to
accumulate in orbits of more negative energy, while others accumulate in
orbits of greater statistical weight. Thus some stars draw closer together,
forming a deeper potential well, while other stars move outwards and may even
escape from the system entirely.
This combination of contraction and expansion takes a number of different
forms. The escape of stars from the cluster can lead to a general contraction
of the remaining system. Heavier stars, as they lose kinetic energy in their
approach to equlpartition, sink toward the cluster center while lighter stars
move outward. The inner isothermal region of a cluster can undergo an
accelerating gravothermal collapse, in which the central core contracts,
losing stars and heating up slightly, while the rest of the cluster expands.
These processes have been investigated wlth detailed computer models, some
following the velocity diffusion process with a Monte-Carlo approach, others
using numerical solutions of the Fokker-Planck equation. For an isolated
cluster these processes seem reasonably well understood.
~Since much of the material presented under this title at the Workshop has
been published in the Proceeding of IAU Symposium 113 (ref. ,)i this paper is
a greatly condensed version.
The gravothermal collapse terminates when the core density becomes high
enough so that binary stars are formed, either by tidal captures in two-body
encounters or directly by three-body encounters. Each binary star tends to
contract when it interacts with passing stars, releasing energy that tends to
terminate the collapse of the core and accelerating the expansion of the outer
regions. To investigate such processes adequately, direct N-body integration
of the equations of motion of the core stars si required, while Monte-Carlo
techniques are applicable to the outer regions.
The evolution of clusters in the post-collapse phase is not yet
thoroughly explored. Once expansion of the inner regions begins it can
continue, powered by binary stars in the core. However, marked gravothermal
oscillations occur under some conditions. The problem is complicated by direct
stellar collisions, which can alter the stellar population in the core,
producing supernovae, black holes and other objects. Since many clusters are
thought to have gone through this collapse phase, an understanding of such
processes is required before detailed models can be compared with real
clusters.
Among areas for possible further research, especially with more powerful
computers, are the following:
)I Detailed effects on cluster evolution resulting from the galactic
gravitational field, which produces a variable field as seen by a cluster.
)2 Analysis of direct collisions between stars and the evolution of the
resulting reaction products, as a result both fo subsequent internal processes
and of further collisions.
)3 Dynamics of the post-collapse phase with realistic assumptions concerning
the anisotropic distribution of stellar velocities and the fate of energy
released by binaries.
)4 Detailed models for overall cluster evolution, beginning with an initial
mass distribution function and taking into account )a early evolution of the
young massive stars, )b perturbations produced by passage of a cluster through
the galactic disc or around the galactic nucleus, )c mass stratification of
stars within a cluster, )d gravothermal collapse, including particularly the
detailed composition of the core at the termination of the collapse phase, )e
the post-collapse phase as affected by the stellar population present.
Reference
.I Dynamics of Star Clusters, IAU Symposium .oN 113, eds: .J Goodman and .P
Hut (Reidel, Dordrecht), 1985, .p 109.
DISC GALAXY DYNAMICS ON THE COMPUTER
J.A. Sellwood
Department of Astronomy
The University
Manchester M13 9PL
Abstract This review a gives feirb summary of the most commonly used techniques
rof disc galaxy simulations and a more a of discussion detailed few numerical seiteltbus
associated with them. The most important these of si gravitational that snoitcaretni
cause the of positions selcitrap to become weakly ,detalerroc increasing the amplitude
of random density .snoitautculf The enhanced noise the causes system to relax more
quickly than would otherwise be expected. tI also has the appearance fluctuating of
larips structure, making ti considerably more tluciffid to demonstrate existence the
of genuine spiral seitilibatsni ni numerical models.
1 Introduction
~¥e have fully to yet comprehend the structure internal of .seixalag Superb new observational
data has taught us that we have only recently begun perceive to the lluf the of extent problems they
present. ,slacitpillE once thought ot be rotationally fattened are objectsj spheroida| now believed
to be ,laixa-irt presenting enormous seitluciffid ni merely constructing an equilibrium model. Disc
galaxies appear to be embedded ni a very massive, but low of halo density elbisivni material. The
uncertainties ni determination of the distribution of mass have done nothing the old simplify to
problems structure spiral of and bar ,ytilibats which llits have no universally accepted .snoitulos
Our stroffe have been spurred on by the hope that a satisfactory understanding internal their of
mechanics give will some clues sa to how galaxies formed. There have been two major of lines
analytical attack: and numerical.
The analytical approach si the more course, of elegant, but that requires generally the problem
be considerably .desilaedi The procedure si tsrif to seek stationary solutions ot the sselnoisilloc
Boltzmann equation ni some assumed deifilpmis mathematical form rof the distribution density and
then to determine their ytilibats ot small amplitude perturbations. This procedure has been been
pursued furthest rof disc ,seixalag but even progress here has been slow and many questions remain
unanswered.
Alternatively, we can yrt simulate to the systems ni the computer. This has two major ad-
arbitrary vantages: nmss snoitubirtsid can be studied at no extra cost and usually calculations the
give some of indication the behaviour. non-linear However, the stluser obtained so raf are llits very
rough and the behaviour si sometimes influenced subtly by the numerical technique.
A close interplay between these two, largely complementary, approaches, can be especially
powerful: experimental results guiding theory, and theory providing standard results against which
to calibrate the codes. Before discussing a few instances of this, I will first outline some of the
techniques used in galaxy simulations. I will focus on the treatment of disc systems and leave a
detailed discussion of spheroidal systems to Prof. van Albada.
2 Summary of Technlques
2.1 Classes of codes
An ideal galaxy simulation code should mimic a eollisionless system with a manageable number
of particles. Attempts to achieve this have branched along two recognisably distinct lines: to expand
in a set of orthogonal functions or to use finite size particles.
Expansion in a set of orthogonal functions is ideal if the mass distribution can be well ap-
proximated by a few members of the basis set. The philosophy here is to use the particles, which
trace the large scale mass distribution in a Monte-Carlo sense, to determine the low-order com-
ponents of the global gravitational field. Equivalently, we can imagine that discarding the higher
order components effectively replaces each particle by a distribution of mass, which is spread in
space as the truncated sum of the basis functions. This automatically suppresses relaxation due to
close encounters. A number of codes based on spherical harmonics have been used for simulations
of spheroidal systems, and will be discussed by van Albada this afternoon. The only disc code to
adopt this approach was devised by Glutton-Brock (1972) who used Hankel-Laguerre functions. He
concluded, however, that the technique could not compete with the efficiency of grid methods when
good spatial resolution was required, as is frequently the case for discs.
The other approach is to sum forces be~,ween particles, either directly or using a grid, and
simply to cut off the inter-particle force at short range. This is usually termed a Finite Size Particle
algorithm, since it implies that a locally confined, usually spherically symmetric, mass cloud is
substituted for each point mass. The short range cut-off can be introduced explicitly through
softening of the force law or implicitly by using a grid - particles within the same grid cell will attract
each other only weakly. Softening is necessary to prevent large angle scattering as two particles
pass, but does little to reduce relaxation from the cumulative effects of long range encounters. This
can be suppressed only by using large numbers of particles.
An apparently convincing demonstration that collisional relaxation is suppressed to a realistic
extent by finite size particles was given by Hohl (1973). Using 100K particles on a 1282 2-D Cartesian
grid, he showed that the time scale for energy equipartition between groups of particles having
different masses was many hundreds of disc rotation periods. However, this test was applied in a
hot, uniformly rotating disc, and it now seems likely that a cool, differentially rotating disc would
have yielded a shorter relaxation time. (See §3.1.)
Most simulation techniques use particles, but it is worth noting that two codes have recentIy
been developed to integrate the coupled collisionless Boltzmann and Poisson equations directly.
Basically, these are fluid dynamical codes in 2-, 4- and (eventually) 6-D phase space. Several results
have already been published by the Japanese group, who use the Cheng-Knorr splitting scheme,
e.g. Nishida et al (1981). Only preliminary results are available from the, perhaps more promising,