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:
(1)
$${\dot{u}}_{n+1}={\dot{u}}_{n}+\frac{1}{2}{h}_{n+1}\left({\ddot{u}}_{n}+{\ddot{u}}_{n+1}\right)$$
(2)
$${u}_{n+1}={u}_{n}+{h}_{n+1}{\dot{u}}_{n}+\frac{1}{2}{h}_{n+1}^{2}{\ddot{u}}_{n}$$
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.
(3)
$${\ddot{u}}_{n}=\frac{{u}_{n+1}2{u}_{n}+{u}_{n1}}{{h}^{2}}$$
From the algorithmic point of view, it is, however, more efficient
to use velocities at half of the time step:
(4)
$${\dot{u}}_{n+\frac{1}{2}}=\dot{u}\left({t}_{n+\frac{1}{2}}\right)=\frac{1}{{h}_{n+1}}\left({u}_{n+1}{u}_{n}\right)$$
so that:
(5)
$${\ddot{u}}_{n}=\frac{1}{{h}_{n+\frac{1}{2}}}\left({\dot{u}}_{n+\frac{1}{2}}{\dot{u}}_{n\frac{1}{2}}\right)$$
(6)
$${h}_{n+\frac{1}{2}}=\left({h}_{n}+{h}_{n+1}\right)/2$$
Time integration is explicit, in that if acceleration
${\ddot{u}}_{n}$
is known (
Combine Modal Reduction), the future velocities and
displacements are calculated from past (known) values in time:

${\dot{u}}_{n+\frac{1}{2}}$
is obtained from Equation 5: (7)
$${\dot{u}}_{n+\frac{1}{2}}={\dot{u}}_{n\frac{1}{2}}+{h}_{n+\frac{1}{2}}{\ddot{u}}_{n}$$
The same formulation is used for rotational velocities.

${u}_{n+1}$
is obtained from Equation 4: (8)
$${u}_{n+1}={u}_{n}+{h}_{n+1}{\dot{u}}_{n+\frac{1}{2}}$$
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.