Central Difference Algorithm
The central difference algorithm corresponds to the Newmark algorithm with $\gamma =\frac{1}{2}$ and $\beta =0$ so that Newarks Method, Equation 7 and Equation 8 become:
with ${h}_{n+1}$ the time step between ${t}_{n}$ and ${t}_{n+1}$ .
It is easy to show that the central difference algorithm ^{1} can be changed to an equivalent form with 3 time steps, if the time step is constant.
From the algorithmic point of view, it is, however, more efficient to use velocities at half of the time step:
so that:

${\dot{u}}_{n+\frac{1}{2}}$
is obtained from Equation 5: $${\dot{u}}_{n+\frac{1}{2}}={\dot{u}}_{n\frac{1}{2}}+{h}_{n+\frac{1}{2}}{\ddot{u}}_{n}$$

${u}_{n+1}$
is obtained from Equation 4: $${u}_{n+1}={u}_{n}+{h}_{n+1}{\dot{u}}_{n+\frac{1}{2}}$$
The accuracy of the scheme is of ${h}^{2}$ order, that is, if the time step is halved, the amount of error in the calculation is one quarter of the original. The time step $h$ may be variable from one cycle to another. It is recalculated after internal forces have been computed.
Analysis of thick and thin shell structures by curved finite elements, Computer Methods in Applied Mechanics and Engineering, 2:419451, 1970.