Analogues of Kahan's method for higher order equations of higher degree

TL;DR

This paper extends Kahan's discretization method to higher order and degree differential equations, introducing explicit, order-preserving schemes that maintain geometric properties and integrability in more complex systems.

Contribution

It proposes a new class of explicit discretization schemes suitable for higher order and degree ODEs, generalizing Kahan's method while preserving key geometric features.

Findings

The new schemes are explicit and order-preserving.

They maintain geometric properties of the original systems.

Applicable to certain higher order, higher degree differential equations.

Abstract

Kahan introduced an explicit method of discretization for systems of first order differential equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method has attracted much interest due to the fact that it preserves many of the geometrical properties of the original continuous system. In particular, a large number of Hamiltonian systems of quadratic vector fields are known for which their Kahan discretization is a discrete integrable system. In this note, we introduce a special class of explicit order-preserving discretization schemes that are appropriate for certain systems of ordinary differential equations of higher order and higher degree.