# Central Difference Algorithm

The central difference algorithm corresponds to the Newmark algorithm with $\gamma =\frac{1}{2}$ and $\beta =0$ so that Newarks Method, 式 7 and 式 8 become:(1)
${\stackrel{˙}{u}}_{n+1}={\stackrel{˙}{u}}_{n}+\frac{1}{2}{h}_{n+1}\left({\stackrel{¨}{u}}_{n}+{\stackrel{¨}{u}}_{n+1}\right)$
(2)
${u}_{n+1}={u}_{n}+{h}_{n+1}{\stackrel{˙}{u}}_{n}+\frac{1}{2}{h}_{n+1}^{2}{\stackrel{¨}{u}}_{n}$

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)
${\stackrel{¨}{u}}_{n}=\frac{{u}_{n+1}-2{u}_{n}+{u}_{n-1}}{{h}^{2}}$
From the algorithmic point of view, it is, however, more efficient to use velocities at half of the time step:(4)
${\stackrel{˙}{u}}_{n+\frac{1}{2}}=\stackrel{˙}{u}\left({t}_{n+\frac{1}{2}}\right)=\frac{1}{{h}_{n+1}}\left({u}_{n+1}-{u}_{n}\right)$
so that:(5)
${\stackrel{¨}{u}}_{n}=\frac{1}{{h}_{n+\frac{1}{2}}}\left({\stackrel{˙}{u}}_{n+\frac{1}{2}}-{\stackrel{˙}{u}}_{n-\frac{1}{2}}\right)$
(6)
${h}_{n+\frac{1}{2}}=\left({h}_{n}+{h}_{n+1}\right)/2$
Time integration is explicit, in that if acceleration ${\stackrel{¨}{u}}_{n}$ is known (Combine Modal Reduction), the future velocities and displacements are calculated from past (known) values in time:
• ${\stackrel{˙}{u}}_{n+\frac{1}{2}}$ is obtained from 式 5: (7)
${\stackrel{˙}{u}}_{n+\frac{1}{2}}={\stackrel{˙}{u}}_{n-\frac{1}{2}}+{h}_{n+\frac{1}{2}}{\stackrel{¨}{u}}_{n}$
The same formulation is used for rotational velocities.
• ${u}_{n+1}$ is obtained from 式 4: (8)
${u}_{n+1}={u}_{n}+{h}_{n+1}{\stackrel{˙}{u}}_{n+\frac{1}{2}}$

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.

1 Ahmad S., Irons B.M., and Zienkiewicz O.C., 「Analysis of thick and thin shell structures by curved finite elements」, Computer Methods in Applied Mechanics and Engineering, 2:419-451, 1970.