Runge{kutta Methods on Manifolds
نویسندگان
چکیده
The subject matter of this paper is the recovery of invariants and conservation laws of ordinary diierential systems by numerical methods. We prove that the most likely candidates for this task, Runge{Kutta schemes, fail to stay on manifolds deened by r-tensors with r 3. As an alternative, we suggest diieomorphically mapping complicated man-ifolds 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 diierential equations (ODEs) y 0 = f(y); t 0; (1.1) is given in the Euclidean space R d and that it is known that for every initial condition y(0) = y 0 its exact solution stays for all t 0 on a manifold M = M(y 0) R d. In other words, y 0 2 M) y(t) 2 M 8 t 0: (1.2) We say that the equation (1:1) is M-invariant.
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