Numerical difficulties with the gravitational n-body problem

Abstract

The gravitational n-body problem is unusually difficult to integrate reliably. In this review, the motion of a set of mass points under their forces of self-gravitation is asserted to be unstable in the sense of a hydrodynamical flow. While this does not imply that an n-body stellar system must collapse or explode, it does mean that we are attempting to treat an unstable system of differential equations by numerical methods. Many of the proofs of convergence or of numerical stability of integration methods are not applicable because they presuppose stability of the differential equations. The history of the problem is reviewed, experimental evidence is given in support of the assertion of instability, and a plea is made for arguments to justify inferences based on numerical experiments even though the differential equations may be unstable.