On Explicit Difference Schemes for Autonomous Systems of Differential Equations on Manifolds

Abstract

The problem of the existence of explicit and at the same time conservative finite difference schemes that approximate a system of ordinary differential equations is investigated. An autonomous system of nonlinear ordinary differential equations on an algebraic manifold V is considered. A difference scheme for solving this system is called conservative, if the calculations of this scheme do not go beyond V, i.e., preserve it exactly. An explicit scheme is understood as such a difference scheme in which a system of linear equations is required to proceed to the next layer. We formulate the problem of constructing an explicit conservative scheme approximating a given autonomous system on a given manifold. For the case of 1-manifold, a solution to this problem is given and geometric obstacles to the existence of such difference schemes are indicated. Namely, it is proved that the scheme exists only if the genus of the integral curve is 1 or 0.