The life-span of backward error analysis for
numerical integrators
E. Hairer
and
Ch. Lubich
Abstract.
Backward error analysis is a useful tool for the study
of numerical approximations to ordinary differential
equations. The numerical solution is formally interpreted
as the exact solution of a perturbed differential equation,
given as a formal and usually divergent series in powers
of the step size.
For a rigorous analysis, this series has to be truncated.
In this article we study the influence of this truncation
to the difference between the numerical solution and the
exact solution of the perturbed differential equation. Results
on the long-time behaviour of numerical
solutions are obtained in this
way. We present applications to the numerical phase portrait near
hyperbolic equilibrium points, to asymptotically stable
periodic orbits and Hopf bifurcation, and to energy conservation
and approximation of invariant tori in Hamiltonian systems.
Key Words.
backward error analysis,
long-term integration, Hamiltonian systems,
Hopf bifurcation