623 |
|
variational methods can capture the decay of energy |
624 |
|
accurately.\cite{Kane2000} Since they are geometrically unstable |
625 |
|
against non-Hamiltonian perturbations, ordinary implicit Runge-Kutta |
626 |
< |
methods are not suitable for Hamiltonian system. Recently, various |
627 |
< |
high-order explicit Runge-Kutta methods \cite{Owren1992,Chen2003} |
628 |
< |
have been developed to overcome this instability. However, due to |
629 |
< |
computational penalty involved in implementing the Runge-Kutta |
630 |
< |
methods, they have not attracted much attention from the Molecular |
631 |
< |
Dynamics community. Instead, splitting methods have been widely |
632 |
< |
accepted since they exploit natural decompositions of the |
633 |
< |
system.\cite{McLachlan1998, Tuckerman1992} |
626 |
> |
methods are not suitable for Hamiltonian |
627 |
> |
system.\cite{Cartwright1992} Recently, various high-order explicit |
628 |
> |
Runge-Kutta methods \cite{Owren1992,Chen2003} have been developed to |
629 |
> |
overcome this instability. However, due to computational penalty |
630 |
> |
involved in implementing the Runge-Kutta methods, they have not |
631 |
> |
attracted much attention from the Molecular Dynamics community. |
632 |
> |
Instead, splitting methods have been widely accepted since they |
633 |
> |
exploit natural decompositions of the system.\cite{McLachlan1998, |
634 |
> |
Tuckerman1992} |
635 |
|
|
636 |
|
\subsubsection{\label{introSection:splittingMethod}\textbf{Splitting Methods}} |
637 |
|
|
654 |
|
problem. If $H_1$ and $H_2$ can be integrated using exact |
655 |
|
propagators $\varphi_1(t)$ and $\varphi_2(t)$, respectively, a |
656 |
|
simple first order expression is then given by the Lie-Trotter |
657 |
< |
formula |
657 |
> |
formula\cite{Trotter1959} |
658 |
|
\begin{equation} |
659 |
|
\varphi _h = \varphi _{1,h} \circ \varphi _{2,h}, |
660 |
|
\label{introEquation:firstOrderSplitting} |