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\section*{Runge-Kutta method}

\textbf{Runge-Kutta approximation schemes} are a family of difference schemes used for iterative numerical solution of [[ordinary differential equation]]s.

\hypertarget{standard_rungekutta_methods}{}\subsubsection*{{Standard Runge-Kutta methods}}\label{standard_rungekutta_methods}

In engineering it is rather standard to use the 4th order Runge-Kutta difference schemes. General overview is at

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods}{Runge-Kutta methods}


\item H. Munthe-Kaas, \emph{High order Runge–Kutta methods on manifolds}, Appl. Num. Math., 29, 115-127 (1999)



\end{itemize}
\hypertarget{for_stochastic_odes}{}\subsubsection*{{For stochastic ODE-s:}}\label{for_stochastic_odes}

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_method_%28SDE%29}{Runge--Kutta method (SDE)}

\end{itemize}
\hypertarget{relation_to_algebraic_problems_lie_theory_and_renormalization}{}\subsubsection*{{Relation to algebraic problems, Lie theory and renormalization}}\label{relation_to_algebraic_problems_lie_theory_and_renormalization}

\begin{itemize}%
\item J.C. Butcher, \emph{Coefficients for the study of Runge-Kutta integration processes}, J. Austral. Math. Soc. \textbf{3} (1963), 185-201

\end{itemize}
In the 1972 Butcher's work Runge-Kutta methods are organized into a group, later called [[Butcher group]]:

\begin{itemize}%
\item J. C. Butcher, An algebraic theory of integration methods, Math. Comp. 26 (1972), 79–106.

\end{itemize}
Brouder has shown a relation of Butcher group to Connes-Kreimer Hopf algebra.

\begin{itemize}%
\item Ch. Brouder, \emph{Runge-Kutta methods and renormalization}, Europ. Phys. J. C12 (2000) 512--534.

\end{itemize}
The Butcher-Connes-Kreimer Hopf algebra is a Hopf subalgebra or Foissy Hopf algebra from work

\begin{itemize}%
\item L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, I.`` Bulletin des Sciences Mathematiques 126, no. 3 (2002): 193–239 (arXiv:math/0105212). MR1905177 ''Les algèbres de Hopf des arbres enracinés décorés. II.`` Bulletin des Sciences Mathematiques 126, no. 4 (2002): 249–88 (arXiv:math/0105212). MR1909461

\end{itemize}
Important family of Lie-algebraic methods for generating integrators are introduced in

\begin{itemize}%
\item P. E. Crouch, R. Grossman, \emph{Numerical integration of ordinary differential equations on manifolds}, J. Nonlinear Sci. \textbf{3}: 1-33 (1993) \href{https://doi.org/10.1007/BF02429858}{doi}

\end{itemize}
\hypertarget{connections_to_symplectic_geometry}{}\subsubsection*{{Connections to symplectic geometry}}\label{connections_to_symplectic_geometry}

See also [[symplectic integrator]]s.

\begin{itemize}%
\item Alberto S. Cattaneo, Benoit Dherin, Giovanni Felder, \emph{Formal symplectic groupoid}, Comm. Math. Phys. \textbf{253} (2005), no. 3, 645--674 \href{http://arxiv.org/abs/math/0312380}{math.SG/0312380} \href{HTTP://dx.doi.org/10.1007/s00220-004-1199-z}{doi}

\end{itemize}
\begin{quote}%
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.


\end{quote}
\begin{itemize}%
\item Robert McLachlan, Klas Modin, Olivier Verdier, \emph{Collective Lie--Poisson integrators on $\mathbb{R}^{3}$}, \href{http://arxiv.org/abs/1307.2387}{arxiv/1307.2387}

\end{itemize}
\begin{quote}%
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 \textbf{symplectic Runge--Kutta schemes}. As a side product, we obtain simple [[symplectic integrator]]s for general Hamiltonian systems on the sphere $S^{2}$.


\end{quote}
\begin{itemize}%
\item J. M. Sanz-Serra, Runge-Kutta Schemes for Hamiltonian Systems, Bit \textbf{28}, pp. 877-883, (1988).

\end{itemize}
[[!redirects Runge-Kutta approximation scheme]]



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