On a property of the Störmer method (Q1571290)

The author considers the system of ordinary differential equations \[ \ddot x=F(x) \] and the \((m+1)\)th-order Stormer method of its numerical solution, \[ x_k=qx_{k-1}-x_{k-2}+{h^2\sum_{i=0}^m}b_iF(x_{k-i}), \tag{1} \] where \(x\) is an \(s\)-dimensional vector, \(F(x)\) is a sufficiently smooth vector function, and \(m\geq 3\). It is shown that discrete dynamical system (1) has a smooth stable invariant manifold on which the multistep Stormer method is reduced to a two-dimensional method. Therefore, the trajectories of the multistep Störmer method asymptotically converge to the trajectories of the two-dimensional method.