Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals

Abstract

Given a quadratic vector field on 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 modified integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a family of integrable maps.

Linearly implicit structure-preserving schemes for Hamiltonian systems

“…One example of linearly implicit methods for Hamiltonian ODEs is Kahan's method, which was designed for solving quadratic ODEs [18] and whose geometric properties have been studied in a series of papers by Celledoni et al [19,20,21]. For Hamiltonian PDEs, Matsuo and Furihata proposed the idea of using multiple points to discretize the variational derivative and thus design linearly implicit energy-preserving schemes [22].…”

How One can Repair Non-integrable Kahan Discretizations. II. A Planar System with Invariant Curves of Degree 6

“…Independently and in the general framework of quadratic ODEs (not necessarily integrable), the bilinear discretization was introduced by Kahan [20]. For some reasons which remain not completely clarified up to now, Kahan's method tends to preserve integrals of motion and integral invariants much more often than any other known general purpose discretization scheme, which was confirmed by extensive studies, see [23][24][25][26][27][28][29][30]32,33,36,40] and [2,3,5,6,38,39]. Generalizations of Kahan's method for higher order ODEs and/or to polynomial vector fields of higher degree were studied in [4,19].…”

How one can repair non-integrable Kahan discretizations

“…Since then, integrability properties of the Kahan's method when applied to integrable systems (also called "Hirota-Kimura method" in this context) were extensively studied, mainly by two groups, in Berlin [13][14][15][16][17][18][19][20][21]24] and in Australia and Norway [2][3][4][5]9,10]. It was demonstrated that, in an amazing number of cases, the method preserves integrability in the sense that the map Φ f (x, ǫ) possesses as many independent integrals of motion as the original system ẋ = f (x).…”