Geometric Integration Part I—Invariants

Abstract

We judge a numerical method by its ability to “approximate” the ODE. It is perfectly natural to

– fix an initial condition,

– fix a time t _f

and ask how closely the method can match x(t _f ), perhaps in the limit h → 0. This led us, in earlier chapters, to the concepts of global error and order of convergence. However, there are other senses in which approximation quality may be studied. We have seen that absolute stability deals with long-time behaviour on linear ODEs, and we have also looked at simple long-time dynamics on nonlinear problems with fixed points. In this chapter and the next we look at another well-defined sense in which the ability of a numerical method to reproduce the behaviour of an ODE can be quantified—we consider ODEs with a conservative nature—that is, certain algebraic quantities remain constant (are conserved) along trajectories. This gives us a taste of a very active research area that has become known as geometric integration, a term that, to the best of our knowledge, was coined by Sanz-Serna in his review article [60]. The material in these two chapters borrows heavily from Hairer et al. [26] and Sanz-Serna and Calvo [61].