Discretization of polynomial vector fields by polarization

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.

[1]  General Approach to Integrating Invertible Dynamical Systems Defined by Transformations from the Cremona group Cr(Pkn) of Birational Transformations , 2000 .

[2]  Matteo Petrera,et al.  On integrability of Hirota-Kimura type discretizations , 2010, 1008.1040.

[3]  A. Pfadler.  Bilinear Discretization of Integrable Quadratic Vector Fields: Algebraic Structure and Algebro-Geometric Solutions , 2011 .

[4]  Y. Suris,et al.  S. V. Kovalevskaya system, its generalization and discretization , 2012, 1208.3726.

[5]  Kinji Kimura,et al.  Discretization of the Lagrange Top , 2000 .

[6]  Abraham OCHOCHE,et al.  General Linear Methods , 2006 .

[7]  V.E.Kravtsov,et al.  Statistics of anomalously localized states at the center of band E=0 in the one-dimensional Anderson localization model , 2012, 1208.4789.

[8]  Jesús María Sanz-Serna,et al.  An unconventional symplectic integrator of W. Kahan , 1994 .

[9]  Y. Suris,et al.  On the Hamiltonian structure of Hirota‐Kimura discretization of the Euler top , 2007, 0707.4382.

[10]  A. Veselov.  Growth and integrability in the dynamics of mappings , 1992 .

[11]  Elena Celledoni,et al.  Geometric properties of Kahan's method , 2012, 1209.1164.

[12]  Andrew Hone,et al.  Non-standard discretization of biological models , 2014, Natural Computing.

[13]  A. Hone,et al.  THREE-DIMENSIONAL DISCRETE SYSTEMS OF HIROTA-KIMURA TYPE AND DEFORMED LIE-POISSON ALGEBRAS , 2008, 0810.5490.

[14]  William Kahan,et al.  Unconventional Schemes for a Class of Ordinary Differential Equations-With Applications to the Korteweg-de Vries Equation , 1997 .

[15]  G. R. W. Quispel,et al.  k-integrals and k-Lie symmetries in discrete dynamical systems , 1996 .

[16]  J. C. Diller,et al.  Dynamics of bimeromorphic maps of surfaces , 2001 .

[17]  P. Santini,et al.  Integrable symplectic maps , 1991 .

[18]  M. P. Bellon,et al.  Algebraic Entropy , 1999 .

[19]  Y. Suris,et al.  Spherical geometry and integrable systems , 2012, 1208.3625.

[20]  Debin Huang.  A coordinate-free reduction for flows on the volume manifold , 2004, Appl. Math. Lett..

[21]  Brynjulf Owren,et al.  A General Framework for Deriving Integral Preserving Numerical Methods for PDEs , 2011, SIAM J. Sci. Comput..

[22]  Elena Celledoni,et al.  Integrability properties of Kahanʼs method , 2014, 1405.3740.

[23]  E. Hairer,et al.  Simulating Hamiltonian dynamics , 2006, Math. Comput..

[24]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .