In this section, the design of a stabilized second-order accurate explicit representation of the form 
(in a nonlinear sense) of the time discretized operator pertaining to the Type 2 classification which inherits unconditional stable features is presented, in which it is not necessary to solve coupled system of equations at every time step except the starting time step (as preliminary efforts; we hope to disseminate more recent advances in the near future).
In fact, if we select 
, 
and m=1, 
in Algorithm 4, a new explicit time discretized operator representation is given by the following:
Since the above representation of two-field d-v form may become somewhat computationally intensive, an alternate d-form is described next.
d-form:
Spectral properties
The curves of spectral radius of the L-stable second-order accurate stabilized explicit time discretized operator representation with different artificial damping ratios are shown in Figure 43 (a - b).
It is evident that: 1) The spectral radii are less than unity for dynamic problems with various physical damping ratio cases; 2) The spectral radius tends to zero as 
tends to infinity, i.e. it is L-stable; 3) The spectral radius is reduced as the artificial damping ratio is increased; and 4) The spectral radius closely mimics the theoretical solution very well for lower frequency.
Figure 43: Spectral radius of second-order accurate explicit representation
Amplitude and phase errors:
These are obtained as follows:
The comparison of dissipation and dispersion characteristics with the central difference method are given in Figures 44(a) and 44(b).
Figure 44: Dissipation and dispersion for second-order accurate explicit representation
Extension of the explicit time discretized representation for nonlinear dynamic systems
A general form of the second-order nonlinear dynamic system is considered as follows:
Accuracy
Following Hoff and Taylor [42], the central difference method is second-order accurate for nonlinear dynamic problems, and hence the suggested method is second-order accurate for nonlinear dynamic problems.
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