I have recently come across the Strang splitting and have some questions. For the differential equation of the form
$$ dy/dt = (L_1 + L_2)y$$
Strang splitting implement the time splitting as
$$ \begin{eqnarray*} \tilde y_1 & = & e^{L_1 \Delta t/2} y_0, & & \bar y_1 = e^{L_2 \Delta t} \tilde y_1, & & y_1 = e^{L_1 \Delta t/2} \bar y_1 \\ \tilde y_2 & = & e^{L_1 \Delta t/2} y_1, & & \bar y_2 = e^{L_2 \Delta t} \tilde y_2, & & y_2 = e^{L_1 \Delta t/2} \bar y_2 \\ ... \\ \tilde y_n & = & e^{L_1 \Delta t/2} y_{n-1}, & & \bar y_n = e^{L_2 \Delta t} \tilde y_n, & & y_n = e^{L_1 \Delta t/2} \bar y_n \\ \end{eqnarray*} $$
However, from the equations above, it is obviously that the half time step with $y_1$ and $\tilde y_2$ can be combined into a single time step, so it is equivalent to:
$$ \begin{eqnarray*} \tilde y_1 & = & e^{L_1 \Delta t/2} y_0, & & \bar y_1 = e^{L_2 \Delta t} \tilde y_1 \\ \tilde y_2 & = & e^{L_1 \Delta t} \bar y_1, & & \bar y_2 = e^{L_2 \Delta t} \tilde y_2 \\ ... \\ \tilde y_n & = & e^{L_1 \Delta t} \bar y_{n-1}, & & \bar y_n = e^{L_2 \Delta t} \tilde y_n \\ y_n & = & e^{L_1 \Delta t/2} \bar y_n \end{eqnarray*} $$
This look like that it is exactly the same as the first order time splitting scheme except the first and last half time step, and the computation is faster with the reduction. Am I missing something here?
Also, what is the time splitting method with four order? How does it look like explicitly? I am looking for some numerical algorithms for solving nonlinear Schroedinger equation.