On the qualitative behaviour of symplectic integrators. I: Perturbed linear systems
This paper deals with the qualitative behaviour of symplectic Runge-Kutta methods in the numerical integration of perturbed linear Hamiltonian systems with dissipative perturbations. The differential systems considered here have the form \({\mathbf x}'= {\mathbf f}^0(x)+ \varepsilon{\mathbf f}^1({\mathbf x}, \varepsilon)\) where \({\mathbf f}^0(x)= {\mathbf {Cx}}\) is a linear Hamiltonian vector field and \(\varepsilon{\mathbf f}^1\) the dissipative perturbation. In the first section the author proves that, describing the system in terms of the action-angle variables of the unperturbed problem \((\varepsilon=0)\), under suitable assumptions the perturbed problem admits an attractive invariant torus \(T_\varepsilon\), for \(\varepsilon\) sufficiently small. Next the existence of an invariant torus \(T_{h,\varepsilon}\) (close to \(T_\varepsilon)\) in the flow defined by a Runge-Kutta method applied to this problem with a fixed stepsize is considered. The remarkable result proved here is that for linearly symplectic methods (or else linearly symmetric) for all \(\varepsilon\in (0,\varepsilon_0)\) the discretized problem has an invariant torus \(T_{h,\varepsilon}\) (close to \(T_\varepsilon\)) for all \(h\leq h_0\) with some positive \(h_0\) independent of the perturbation parameter \(\varepsilon\), while for non symplectic methods \(h_0= h_0 (\varepsilon)\) depends on \(\varepsilon\), in a way that the local discretization error of the method must be \(\leq\varepsilon\). Thus, in the case of non symplectic methods to ensure the existence of these tori for \(\varepsilon\to 0\) a simultaneous restriction on the maximum stepsize must be imposed. This fact shows the favourable behaviour of symplectic Runge-Kutta methods in order to preserve the qualitative properties of the flow in the above systems.
scientific article; zbMATH DE number 1096028