Geometric properties of Kahan's method

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.

Revised version

Cited by in corpus (39)

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• Conservative methods for dynamical systems
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• New classes of quadratic vector fields admitting integral-preserving Kahan-Hirota-Kimura discretizations
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• Reduced order modelling of nonlinear cross-diffusion systems
• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• Discretized Fast-Slow Systems with Canards in Two Dimensions
• A three-dimensional generalization of QRT maps
• On the singularity structure of Kahan discretizations of a class of quadratic vector fields
• Why geometric integration?
• An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps
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• New results on integrability of the Kahan-Hirota-Kimura discretizations
• A construction of a large family of commuting pairs of integrable symplectic birational 4-dimensional maps
• A new approach to integrals of discretizations by polarization
• How one can repair non-integrable Kahan discretizations. II. A planar system with invariant curves of degree 6
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• Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
• Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation
• Geometric and integrability properties of Kahan's method
• A construction of commuting systems of integrable symplectic birational maps
• Geometric integration of non-autonomous Hamiltonian problems
• Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. II. Systems with a linear Poisson tensor
• A construction of commuting systems of integrable symplectic birational maps. Lie-Poisson case
• Preservation of the invariants of Lotka-Volterra equations by iterated deferred correction methods
• Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian
• Kahan-Hirota-Kimura maps preserving original cubic hamiltonians
• Two regularized energy-preserving finite difference methods for the logarithmic Klein-Gordon equation