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Abstract : |
Abstract. 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. 1 Numerical Methods and Differentiable Manifolds This paper addresses itself to a topic of an increasing interest in the numerical community, the discretization of ordinary differential equations (ODEs) on differentiable manifolds. Suppose that it is known that the solution of the ODE system y, |
