| 846 |
|
\] |
| 847 |
|
Applying Baker-Campbell-Hausdorff formula to Sprang splitting, we |
| 848 |
|
can obtain |
| 849 |
< |
\begin{eqnarray} |
| 849 |
> |
\begin{eqnarray*} |
| 850 |
|
\exp (h X/2)\exp (h Y)\exp (h X/2) & = & \exp (h X + h Y + h^2 [X,Y]/4 + h^2 [Y,X]/4 \\ |
| 851 |
|
& & \mbox{} + h^2 [X,X]/8 + h^2 [Y,Y]/8 \\ |
| 852 |
< |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24\\ |
| 853 |
< |
& & \mbox{} + \ldots ) |
| 854 |
< |
\end{eqnarrary} |
| 852 |
> |
& & \mbox{} + h^3 [Y,[Y,X]]/12 - h^3[X,[X,Y]]/24 + \ldots ) |
| 853 |
> |
\end{eqnarray*} |
| 854 |
|
Since \[ [X,Y] + [Y,X] = 0\] and \[ [X,X] = 0\], the dominant local |
| 855 |
|
error of Spring splitting is proportional to $h^3$. The same |
| 856 |
|
procedure can be applied to general splitting, of the form |
| 858 |
|
\varphi _{b_m h}^2 \circ \varphi _{a_m h}^1 \circ \varphi _{b_{m - |
| 859 |
|
1} h}^2 \circ \ldots \circ \varphi _{a_1 h}^1 . |
| 860 |
|
\end{equation} |
| 861 |
< |
Careful choice of coefficient $a_1 ,\ldot , b_m$ will lead to higher |
| 861 |
> |
Careful choice of coefficient $a_1 \ldot b_m$ will lead to higher |
| 862 |
|
order method. Yoshida proposed an elegant way to compose higher |
| 863 |
|
order methods based on symmetric splitting. Given a symmetric second |
| 864 |
|
order base method $ \varphi _h^{(2)} $, a fourth-order symmetric |
