Geometric and integrability properties of Kahan's method
Provides a theoretical foundation for the integrability of Kahan discretizations, benefiting researchers in numerical analysis and dynamical systems.
The paper proves that Kahan's discretization method exactly preserves a nearby modified integral for quadratic vector fields with a quadratic first integral depending on two variables, and uses this to construct a new 10-parameter family of integrable maps from integrable quadratic vector fields.
Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discretization method exactly preserves a nearby modifed integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a novel 10-parameter family of integrable maps.