what is name of this numerical scheme for ode? Let's have system of ODEs
$$
\dot x(t) = A(t)x(t)
$$
I came up with this numerical scheme:
$$
x_{n+1} = e^{\frac{h}{2}A(t_{n+1})}e^{\frac{h}{2}A(t_n)}x_n
$$
where $h$ is time step, $t_n = nh$ and $x_n$ is approximate value of $x(t_n)$.
It comes from idea that at each time $t_n$ I freeze time in matrix $A$ and move along solution of $\dot x(t) = A(t_n)x(t)$. From this I get scheme:
$$
x_{n+1} = e^{hA(t_n)}x_n.
$$
To make it time reversible I modified it to the form already mentioned. 
There has to be a theory of this kind of schemes so I would like to know its name so I can search for literature a find out more about these schemes.
 A: I would call it a Magnus integrator, as it is related to the Magnus expansion (see Wikipedia article). You can read a bit more about these methods in Section 5 of: 
S. Blanes, F. Casas, J.A. Oteo, J. Ros (2009). "The Magnus expansion and some of its applications". Phys. Rep. 470 (5-6): 151–238; arxiv:0810:5488. 
Normally, the second-order Magnus integrator is taken to be 
$$
x_{n+1} = e^{hA(t_{n+1/2})} x_n \quad\text{where}\quad t_{n+1/2} = t_n + \frac12 h.
$$ 
This is time-reversible and second-order as the method you mention, but probably a bit simpler.
A: This is essentially just a two-step solution.
Consider $x_{m+1} = e^{h'A(t_m)}x_m$, where $h' = \frac{h}{2}$, and $m$ is the index for your "half-step."
This is in essence the solution to the ode $x' = Ax$ at $t_m$.
If we take another half-step, then we have $$x_{m+2} = e^{h'A(t_{m+1})}x_{m+1} = e^{h'A(t_{m+1})}e^{h'A(t_m)}x_m.$$
This is almost exactly what you have, except we have the matrix evaluated at the half-step and not the full step.
Effectively, what you have is the predictor step of a PECE (Predict, evaluate, correct, evaluate) method. Unfortunately, I don't forsee this method having nice convergence properties without a corrector step. The "time reversibility" you talk about is essentially what a PECE method does. It looks forward, then adjusts that prediction by "looking backward" and using, say, the trapezoid rule to improve that prediction.
