Discretization of polynomial vector fields by polarization

TL;DR

This paper generalizes Kahan's quadratic vector field integration method to cubic and higher degree polynomial vector fields, demonstrating that the new discretizations preserve modified measure and energy for polynomial Hamiltonian systems.

Contribution

The paper extends Kahan's method to higher degree polynomial vector fields and proves measure and energy preservation in these cases.

Findings

Preserves modified measure and energy for cubic polynomial Hamiltonian vector fields.

Generalizes Kahan's method beyond quadratic vector fields.

Applicable to higher degree polynomial Hamiltonian systems.

Abstract

A novel integration method for quadratic vector fields was introduced by Kahan in 1993. Subsequently, it was shown that Kahan's method preserves a (modified) measure and energy when applied to quadratic Hamiltonian vector fields. Here we generalize Kahan's method to cubic resp. higher degree polynomial vector fields and show that the resulting discretization also preserves modified versions of the measure and energy when applied to cubic resp. higher degree polynomial Hamiltonian vector fields.