Discretization of polynomial vector fields by polarization

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

Numerical discretization of completely integrable nonlinear Hamiltonian systems

“…This behavior for the LV system is explained in [20]. Further, more general properties of the method are explained in [12]; restricted to quadratic fields it was found to be an iteration of RK type, and for cubic systems the Kahan A generalization was later found for k-homogeneous polynomial Hamiltonians [14], which gives a k-step method. A function fðx 1 ; …; x n Þ is said to be homogeneous of degree d, or d-homogeneous if fðλx 1 ; …; λx n Þ ¼ λ d pðx 1 ; …; x n Þ for any scalar λ.…”

“…Discovered by chance, Kahan's method [11] was shown to preserve integrability for polynomial Hamiltonians in the plane, where the polynomial has maximum degree three [12,13]. It might be possible to extend these results by polarization to homogeneous polynomial Hamiltonians of higher degree [14]. No general result is available even for polynomial Hamiltonians, or dimensions larger than two, not to mention the more complicated Hamiltonians that arise from the motion of charged particles in arbitrary electromagnetic fields.…”

Linearly implicit structure-preserving schemes for Hamiltonian systems

“…is the symmetric bilinear form obtained by polarisation of the quadratic formQ [19]. Polarisation, which maps a homogeneous polynomial function to a symmetric multi-linear form in more variables, was used to generalise Kahan's method to higher degree polynomial vector fields in [29]. Suppose we restrict the problem (2.1) to be a Hamiltonian system on a Poisson vector space with a constant Poisson structure:ẏ = A∇H (y),…”

Analogues of Kahan’s Method for Higher Order Equations of Higher Degree

“…In the context of Lotka-Volterra models, a variant of Kahan's method with similar properties was discovered by Mickens [19], who had previously considered various examples of nonstandard discretization methods [18], but a more rapid growth of interest in Kahan's method began when Hirota and Kimura independently proposed the rules (2) for the discretization of the Euler equations for rigid body motion, finding that the resulting discrete system is also completely integrable [11], and this has led to the search for other discrete integrable systems arising in this way [12], with a survey of several results given in [20], and some more recent examples in [21] and [22], for instance. Many of the geometrical properties of Kahan's method for quadratic vector fields are based on the polarization identity for quadratic forms [3], and recently this has led to a generalization of Kahan's method that can cope with vector fields of degree three or more, by using higher degree analogues of polarization [4]. One disadvantage of the latter method for higher degree vector fields is that, in common with multistep methods in numerical analysis, one must use extra grid points for the discretization, so the original ODE system does not provide enough initial values to start the iteration of the discrete version.…”