Newmark's method is a one step integration method. The state of the system at a given time
${t}_{n+1}={t}_{n}+h$
is computed using Taylor's formula:
(1)
$$f\left({t}_{n}+h\right)=f\left({t}_{n}\right)+h{f}^{\prime}\left({t}_{n}\right)+\frac{{h}^{2}}{2}{f}^{\left(2\right)}\left({t}_{n}\right)+\mathrm{...}+\frac{{h}^{s}}{s!}{f}^{\left(s\right)}\left({t}_{n}\right)+{R}_{s}$$
(2)
$${R}_{s}=\frac{1}{s!}{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n}+h}{\int}}{f}^{\left(s+1\right)}\left(\tau \right){\left[{t}_{n}+h\tau \right]}^{s}d\tau}$$
The preceding formula allows the computation of displacements and velocities of the system
at time
${t}_{n+1}$
:
(3)
$${\dot{u}}_{n+1}={\dot{u}}_{n}+{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\ddot{u}\left(\tau \right)d\tau}$$
(4)
$${u}_{n+1}={u}_{n}+h{\dot{u}}_{n}+{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\left({t}_{n+1}\tau \right)\ddot{u}\left(\tau \right)d\tau}$$
The approximation consists in computing the integrals for acceleration in
Equation 3 and in
Equation 4 by numerical
quadrature:
(5)
$$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\ddot{u}\left(\tau \right)d\tau}=\left(1\gamma \right)h{\ddot{u}}_{n}+\gamma h{\ddot{u}}_{n+1}+{r}_{n$$
(6)
$$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\left({t}_{n+1}\tau \right)\ddot{u}\left(\tau \right)d\tau}=\left(\frac{1}{2}\beta \right){h}^{2}{\ddot{u}}_{n}+\beta {h}^{2}{\ddot{u}}_{n+1}+{{r}^{\prime}}_{n$$
According to the values of
$\gamma $
and
$\beta $
, different algorithms can be derived:

$\gamma =0,\beta =0$
: pure explicit algorithm. It can be shown that it is
always unstable. An integration scheme is stable if a critical time step exists so that,
for a value of the time step lower or equal to this critical value, a finite
perturbation at a given time does not lead to a growing modification at future time
steps.

$\gamma =1/2,\beta =0$
: central difference algorithm. It can be shown that it is
conditionally stable.

$\gamma =1/2,\beta =1/2$
: Fox & Goodwin algorithm.

$\gamma =1/2,\beta =1/6$
: linear acceleration.

$\gamma =1/2,\beta =1/4$
: mean acceleration. This integration scheme is the
unconditionally stable algorithm of maximum accuracy.