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.