Runge-Kutta approximation schemes are a family of difference schemes used for iterative numerical solution of ordinary differential equations.
In engineering it is rather standard to use the 4th order Runge-Kutta difference schemes. General overview is at
wikipedia Runge-Kutta methods
H. Munthe-Kaas, High order Runge–Kutta methods on manifolds, Appl. Num. Math., 29, 115-127 (1999)
In the 1972 Butcher’s work Runge-Kutta methods are organized into a group, later called Butcher group:
Brouder has shown a relation of Butcher group to Connes-Kreimer Hopf algebra.
The Butcher-Connes-Kreimer Hopf algebra is a Hopf subalgebra or Foissy Hopf algebra from work
Important family of Lie-algebraic methods for generating integrators are introduced in
See also symplectic integrators.
The multiplicative structure of the trivial symplectic groupoid over $\mathbb{R}^d$ associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevich’s deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge–Kutta theory of the numeric integration of ODE’s.
We develop Lie–Poisson integrators for general Hamiltonian systems on $\mathbb{R}^{3}$ equipped with the rigid body bracket. The method uses symplectic realisation of $\mathbb{R}^{3}$ on $T^{*}\mathbb{R}^{2}$ and application of symplectic Runge–Kutta schemes. As a side product, we obtain simple symplectic integrators for general Hamiltonian systems on the sphere $S^{2}$.
Last revised on April 3, 2018 at 15:06:15. See the history of this page for a list of all contributions to it.