| 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} |
| 752 |
|
|
| 753 |
|
The Baker-Campbell-Hausdorff formula\cite{Gilmore1974} can be used |
| 754 |
|
to determine the local error of a splitting method in terms of the |
| 755 |
< |
commutator of the |
| 756 |
< |
operators(Eq.~\ref{introEquation:exponentialOperator}) associated |
| 757 |
< |
with the sub-propagator. For operators $hX$ and $hY$ which are |
| 757 |
< |
associated with $\varphi_1(t)$ and $\varphi_2(t)$ respectively , we |
| 758 |
< |
have |
| 755 |
> |
commutator of the operators associated with the sub-propagator. For |
| 756 |
> |
operators $hX$ and $hY$ which are associated with $\varphi_1(t)$ and |
| 757 |
> |
$\varphi_2(t)$ respectively , we have |
| 758 |
|
\begin{equation} |
| 759 |
|
\exp (hX + hY) = \exp (hZ) |
| 760 |
|
\end{equation} |
