Geometric properties of Kahan's method
TLDR
In this paper, it was shown that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method, which produces large classes of integrable rational mappings in two and three dimensions.Abstract:
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 map determined by this discretization has a conserved modified Hamiltonian and an invariant measure, a combination previously unknown amongst Runge–Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known.read more
Citations
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Journal ArticleDOI
Integrability properties of Kahan's method
TL;DR: In this article, the authors present several integrable quadratic vector fields for which Kahan's discretization method preserves integrability, including generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painleve equation.
Journal ArticleDOI
Integrability properties of Kahanʼs method
TL;DR: In this article, the authors present several integrable quadratic vector fields for which Kahan's discretization method preserves integrability, including generalized Suslov and Ishii systems, Nambu systems, Riccati systems, and the first Painleve equation.
Journal ArticleDOI
Structure-preserving Runge-Kutta methods for stochastic Hamiltonian equations with additive noise
Pamela Burrage,Kevin Burrage +1 more
TL;DR: It is shown that there are classes of Runge-Kutta methods that are very effective in preserving the expectation of the Hamiltonian, but care has to be taken in how the Wiener increments are sampled at each timestep.
Journal ArticleDOI
Discretization of polynomial vector fields by polarization
TL;DR: In this article, it was shown that Kahan's method preserves a modified measure and energy when applied to quadratic vector fields, and that it can be applied to any vector field.
Journal ArticleDOI
Novel high-order energy-preserving diagonally implicit Runge–Kutta schemes for nonlinear Hamiltonian ODEs
Hong Zhang,Xu Qian,Songhe Song +2 more
TL;DR: The invariant energy quadratization approach is utilized to transform the nonlinear Hamiltonian ODE into an equivalent form which has a quadratic energy and comparisons with the symplectic Runge–Kutta scheme and energy-preserving average vector field method are presented to demonstrate the energy- Preserving property and accuracy of the proposed method.
References
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Book
Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
TL;DR: In this article, the authors present a model for symmetric integration of non-Canonical Hamiltonian systems and a model of symmetric Hamiltonian integration with symmetric integrators.
Journal ArticleDOI
Integrable symplectic maps
TL;DR: In this paper, the authors give a terse survey of symplectic maps, their canonical formulation and integrability, and introduce a rigorous procedure to construct integrable symplectic map starting from integrably evolution equations on lattices.
BookDOI
Discrete Integrable Systems
TL;DR: In this paper, the authors give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based, assuming Theorem 3.7.
Journal ArticleDOI
Energy-preserving Runge-Kutta methods
Elena Celledoni,Robert I. McLachlan,David I. McLaren,Brynjulf Owren,G. Reinout W. Quispel,W. M. Wright +5 more
TL;DR: It is shown that while Runge-Kutta methods cannot preserve polynomial invariants in gen- eral, they can preserved polynomials that are the energy invariant of canonical Hamiltonian systems.
Book
Discrete Integrable Systems: QRT Maps and Elliptic Surfaces
TL;DR: The QRT Map as discussed by the authors is the pencil of biquadratic curves in the projective plane of the QRT surface and is used to measure the distance between two points.