On unconventional discretisations

We present various discretisation techniques that mark a departure from the classical algorithms of numerical analysis, in the aim of preserving the physical behaviour of the solutions to the differential systems we are discretising, even for discretisation steps that are not very small.

These techniques are inspired by the principles laid down by Mickens and are close to those proposed by Hirota and Kahan.

Detailed applications of these methods are presented for the logistic equation and for the Lotka-Volterra system.

We discuss the importance of preserving positivity whenever the latter is an expected property of the solution and give a practical rule for achieving this.

Possible problems arising from the discretisations we propose are also discussed.

A section is devoted to discretisations of integrable systems aimed at preserving integrability.

Finally we conclude this review with a discussion of the possible relevance of a discrete space-time.