Related papers: A new approach to integrals of discretizations by …

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…

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…

We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretization, using Darboux Polynomials (DPs). Somewhat similar…

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.…

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…

Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.

We present some new families of quadratic vector fields, not necessarily integrable, for which their Kahan-Hirota-Kimura discretization exhibits the preservation of some of the characterizing features of the underlying continuous systems…

We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtained in this way can be geometrically understood as the…

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic Hamiltonian vector field with a linear Poisson tensor and with a…

We consider an Ansatz for the study of the existence of formal integrals of motion for Kahan-Hirota-Kimura discretizations. In this context, we give a combinatorial proof of the formula of Celledoni-McLachlan-Owren-Quispel for an integral…

We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the…

Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the…

Kahan discretization is applicable to any system of ordinary differential equations on $\mathbb R^n$ with a quadratic vector field, $\dot{x}=f(x)=Q(x)+Bx+c$, and produces a birational map $x\mapsto \widetilde{x}$ according to the formula…

We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line,…

The main result of this paper is the discretization of Hamiltonian systems of the form $\ddot x = -K \nabla W(x)$, where $K$ is a constant symmetric matrix and $W\colon\mathbb{R}^n\to \mathbb{R}$ is a polynomial of degree $d\le 4$ in any…

We discuss the singularity structure of Kahan discretizations of a class of quadratric vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an…

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector…

We find a novel one-parameter family of integrable quadratic Cremona maps of the plane preserving a pencil of curves of degree 6 and of genus 1. They turn out to serve as Kahan-type discretizations of a novel family of quadratic vector…

We introduce a novel technique for constructing higher-order variational integrators for Hamiltonian systems of ODEs. In particular, we are concerned with generating globally smooth approximations to solutions of a Hamiltonian system. Our…

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility…