RUNGE{KUTTA METHODS ON MANIFOLDS

The subject matter of this paper is the recovery of invariants and conservation laws of ordinary di erential systems by numerical methods. We prove that the most likely candidates for this task, Runge{Kutta schemes, fail to stay on manifolds de ned by r-tensors with r 3. As an alternative, we suggest di eomorphically mapping complicated manifolds to simpler ones. This procedure allows for recovery of invariants that are intractable in a classical setting and it emphasizes the crucial role of the topology of underlying manifolds. 1. Invariants and numerical methods Let us suppose that an autonomuos system of ordinary di erential equations (ODEs) y0 = f (y); t 0; (1.1) is given in the Euclidean space R and that it is known that for every initial condition y(0) = y0 its exact solution stays for all t 0 on a manifoldM =M(y0) R . In other words, y0 2 M ) y(t) 2 M 8 t 0: (1.2) We say that the equation (1:1) is M-invariant. M-invariance often conveys an important information about the underlying di erential system, for example the satisfaction of a conservation law, an integral, etc. It is usually desirable and often essential to retain this feature under discretization. Classical approach to numericalmethods for ODEs pays little attention to the issue of invariance, the emphasis being on deriving accuracy within a speci ed error tolerance and on the recovery of correct asymptotic behaviour of the solution. This is perfectly satisfactory in many situations and we have no intention to denigrate the impressive body of contemporary work in numerical solution of ODEs. However, there exists a wide range of problems that require the exact retention of invariants, e.g. Hamiltonian systems and isospectral ows, whereby the classical approach falls short of our requirements. We refer the reader to (Iserles and Zanna, 1995), where this issue is debated at a greater length. Recent years have witnessed a growing amount of attention devoted to numerical retention of invariance. Much of the work has been specialized to speci c manifolds: the symplectic manifold for Hamiltonian equations (Sanz-Serna and Calvo, 1994), the unitary group O(d) for unitary ows (Dieci, Russell and Van Vleck, 1994) and the isospectral manifold for Toda lattice equations, double-bracket equations and other isospectral ows (Calvo, Iserles and Zanna, 1995). However, outlines of a more general theory have started to emerge lately, mainly in the work of Peter Crouch and R. Grossman (1993) and Hans Munthe-Kaas (1995). The aim of the present paper is to pursue the general theme, to understand how well can existing methods retain M-invariance for di erent manifoldsM and what can be done to improve their e cacy. Section 2 expands upon a theme to which we have already addressed ourselves in a somewhat lesser generality in (Calvo et al., 1995), namely manifolds that are de ned as a level set of a r-tensor, r 2 N. The case r = 1 is trivial, while r = 2 has been already determined by Cooper (1987). We prove that no noncon uent Runge{Kutta method can retain M-invariance for all manifolds described by a symmetric r-tensor whenever r 3. In particular, no Runge{Kutta method can stay on the manifold