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\title{\bf This is the title of our paper\footnote{This work was 
supported by CMUC and HCM project ERBCHRXCT930407.}}
\author{
     First Author$^{a,}$\thanks{Corresponding author. E-mail: fauthor@mat.uc.pt.}, 
	 Second Author$^{b}$ \\ \ \\
	 {\small \it  $^{a}$ Departamento de Matem\'atica da 
	 Universidade de Coimbra, 3000 Coimbra, Portugal} \\ 
	 {\small \it  $^{b}$ Departamento de Engenharia Qu\'\i mica 
	 da Universidade de Coimbra, Largo Marqu\^es de Pombal,} \\ 
	 {\small \it 3000 Coimbra, Portugal}
	 }
\date{}
\maketitle
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\begin{abstract}
	We consider numerical methods that integrate differential systems 
	$dy/dt=f(y)$ of Hamiltonian type by taking advantage of a decomposition of 
	the right-hand side $f=\sum f^{[\nu]}$.
\end{abstract}

{\it Keywords:} Additive Runge-Kutta method; Symplectic integration; Splitting
\medskip

{\it AMS classification:} 65L05 
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%  Introduction
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\section{Introduction}
Most numerical methods for the integration of 
systems of differential equations
\begin{equation}
	\label{ode}
    \frac{dy}{dt}=f(y)
\end{equation}
use the right-hand side $f$ only through a subroutine that finds 
$f(y)$ at any given point $y$. Other methods require, in addition, 
subroutines for the evaluation of the Jacobian $f'(y)$ or of higher 
derivatives of $f$. All these methods, by ignoring all details of the 
structure of $f$, may provide general-purpose, black-box library 
integrators. 

However, in many applications $f$ possesses, in a 
natural way, a decomposition 
\begin{equation}
	\label{decomp}
	f(y)=\sum_{\nu=1}^N f^{[\nu]}(y).
\end{equation}
and one may wish to consider methods that take advantage of the 
structure of $f$. 
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\newpage

\centerline{\sc Equation layout}
\vspace{1cm}

Assume that (\ref{ode}) is a Hamiltonian system, i.e. that $D$ is 
even $D=2d$ and that there exists a real-valued function $H(y)$ such that
\begin{equation}
	\label{ham} 
	f(y)=J^{-1}\frac{\partial H}{\partial y}(y), 
\end{equation}
where $\partial H/\partial y= (\partial H/\partial 
y^1,\ldots, \partial H/\partial y^{2d})^T$ and $J$ is the $2d\times 2d$ matrix
\[
J=\left [ \begin{array}{cc}
			0_d  & I_d \\
			-I_d & 0_d
	       \end{array} 
  \right ].
\]
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\vspace{1.5cm}
\centerline{\sc Reference layout}
\vspace{1cm}
References should be ordered alphabetically and should be conformal 
with the following example for papers \cite{ARW95}, books 
\cite{HNW93} and contributed volumes \cite{San92}.

\begin{thebibliography}{3}
	\bibitem{ARW95} U.M. Ascher, S.J. Ruuth, B.T.R. Wetton, 
	Implicit-explicit methods for time-dependent 
	partial differential equations, {\sl SIAM J. Numer. Anal.} {\bf 32} (1995) 
	797--823.
	
	\bibitem{HNW93} E. Hairer, S.P. N\o rsett, G. Wanner, 
	{\sl Solving Ordinary Differential Equations I, Nonstiff Problems}, 
	(2nd ed.), Springer-Verlag, Berlin, 1993.
	
	\bibitem{San92} J.M. Sanz-Serna, On difference schemes and 
	symplectic geometry, in: D. Broomhead, A. Iserles (Eds.), 
	{\sl Dynamics of Numerics and Numerics of Dynamics}, Oxford University 
	Press, Bristol, 1992, pp. 81--106.
\end{thebibliography}

In text, references should be made by giving the reference number 
within brackets.
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