Stabilization of invariants of discretized differential systems

Many problems of practical interest can be modeled by differential systems where the solution lies on an invariant manifold defined explicitly by algebraic equations. In computer simulations, it is often important to take into account the invariant's information, either in order to improve upon the stability of the discretization (which is especially important in cases of long time integration) or because a more precise conservation of the invariant is needed for the given application. In this paper we review and discuss methods for stabilizing such an invariant. Particular attention is paid to post-stabilization techniques, where the stabilization steps are applied to the discretized differential system. We summarize theoretical convergence results for these methods and describe the application of this technique to multibody systems with holonomic constraints. We then briefly consider collocation methods which automatically satisfy certain, relatively simple invariants. Finally, we consider an example of a very long time integration and the effect of the loss of symplecticity and time-reversibility by the stabilization techniques. This revised version was published online in June 2006 with corrections to the Cover Date.