Geometric Integration Part I—Invariants

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].