Numerical Methods on (and off) Manifolds

It is often known from theoretical analysis that the exact solution of an ordinary differential system lies on a specific differentiable manifold and there are important advantages in retaining this feature under discretization. In this paper we examine whether the correct manifold is retained by a class of discretization methods that includes explicit multistep and multiderivative schemes. We obtain a necessary and sufficient condition for the retention of invariance under discretization. In particular, we prove that no such method can be expected to stay on a quadratic manifold. More specifically, given such a method and an arbitrary quadratic manifold N, there exists an ordinary differential equation whose exact solution lies on N, yet its discretization by the underlying method departs from the manifold.