A gravitational *N*-body system
consists of a set of *N* particles of mass , and position
, *i*=1,..*N*, all moving under the force of their mutual
gravitational attraction. The force particle *j* exerts on particle
*i* is

where *G* is Newton's gravitational constant, ,
, and . The total
force on particle *i* is the sum of the forces exerted on it by all
the other particles.

We study the reliability of direct numerical simulation of the large
*N* gravitational *N*-body problem.
A partial list of errors
introduced in such simulations includes [33]: finite-*N*
sampling, or *discreteness noise*, because the *N* used in
simulations usually is much smaller than the *N* of the system being
modelled; *force softening*, which eliminates the singularity at
*r*=0 and lessens spurious collisional effects introduced by finite-*N*
sampling [9];
using fast but *approximate force algorithms* such as
tree codes [6, 45] or particle-mesh methods
(*eg., * [61]);
numerical ordinary differential equation (ODE) integration
*truncation* error; and *machine roundoff* error. The goal of
the study of reliability is to ascertain what effect these *input*
or *local* errors have on the *output* or *global* error.
Less stringently, we can be interested in the effect of these errors
on statistical measurements taken from the simulations.

What is meant by ``reliability'' will certainly depend on what we are
trying to measure or produce with our simulations. For example, the
goal of a simulation of a cold, collisionless disk may be simply to
reproduce global effects like spiral arms, or to test the effect of
various distributions of dark matter. In such simulations, the
microscopic dynamics of individual particles are of little interest
as long as the global structure is valid. In contrast, when simulating
a highly collisional globular cluster, it is widely known (*eg., *
[75]) that microscopic effects, like close encounters and
formation of binary stars in the core, have significant impact on the
global evolution of the cluster. Thus, to reliably reproduce global
properties of globular clusters, these microscopic details must be
taken into account, even though the particular happenings of any
individual simulation may be of little interest.

We are interested both in the theory and practice of error control.
Although we realize that a practical error measure must be devised in order
to control error, we are also interested in the theoretical question of
*what is the strongest statement that we can make about how a
numerical simulation reflects the real system?* A very strong
statement, for example, might have something to do with shadowing of
phase-space trajectories, whereas for practical error control we may
attempt to conserve energy to some given accuracy. We would be
particularly interested in a statement that would link, for example,
the quality of energy conservation to the shadowing property.

The present paper is a brief tour of some of the literature concerning
the topic of the title. The survey is by no means
complete. We will look at some
simulations done by astronomers, and the kinds of measurements taken.
We will then connect *N*-body simulation to the study of chaos, because
the *N*-body problem has *sensitive dependence on initial
conditions*, and to the study of *shadowing*, which is one of many
kinds of global error analysis techniques for ODE integration.
Finally, we will briefly mention the connection between shadowing and
Boundary Value Problems for ODEs.

Fri Dec 27 17:41:39 EST 1996

Access count (updated once a day) since 1 Jan 1997: 8550