Numerical flow-box theorems under structural assumptions (Q2748867)

The aim of the reviewed article is to continue the investigation of \textit{O. Gonzalez, D. J. Higham} and \textit{A. M. Stuart} [IMA J. Numer. Anal. 19, No. 2, 169-190 (1999; Zbl 0935.34003)] and to present similar results for the discretized (perfectly modified) equation \(\dot x= f_{h,x_{0}}(x)\) in the vicinity of an arbitrary nonequilibrium point \(x_0\) of the original equation \(\dot x=f(x)\). It is known that the discretized solutions of an autonomous ordinary differential equation are the exact solutions of a modified equation: for a stepsize \(h\) sufficiently small the original discretization operator is the time-\(h\) map of the solution operator of the modified equation. Here this result is extended to the following categories of differential equations and discretizations: \(1^0.\) Preserving a finite number of first integrals, \(2^0.\) Preserving the volume form, \(3^0.\) Preserving the canonical symplectic form.