STABILIZATION OF DAES AND INVARIANT-MANIFOLDS

Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve co nstraint differentiation and problem stabilization, thus obtaining a s tabilized index reduction. A popular method is Baumgarte stabilization , but the choice of parameters to make it robust is unclear in practic e. Here we explain why the Baumgarte method may run into trouble. We t hen show how to improve it. We further develop a unifying theory for s tabilization methods which includes many of the various techniques pro posed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a si mple way, generally different from the ODE discretization, and (iii) u se orthogonal projections whenever possible. The best methods thus obt ained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions.