Geometric properties of Kahan's method

TL;DR

This paper reveals that Kahan's method for quadratic vector fields is equivalent to a Runge--Kutta method and demonstrates its ability to preserve modified Hamiltonians and invariant measures in Hamiltonian systems, leading to integrable mappings.

Contribution

It establishes the equivalence of Kahan's method to a Runge--Kutta method and uncovers its geometric properties in Hamiltonian systems, including conservation laws.

Findings

Kahan's method is equivalent to a Runge--Kutta method.

The method preserves a modified Hamiltonian in Hamiltonian systems.

It produces integrable rational mappings in low dimensions.

Abstract

We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge--Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge--Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.