## 7 Numerical Properties of CVM
### Overshooting  
[PERSON] and [PERSON] found that an integration algorithm has overshooting behavior. Later, [PERSON] and [PERSON] proposed a technique to identify such an overshoot. To determine the overshoot behavior of an integration algorithm, one can compute the velocity and discrete displacement in terms of the previous step data. In general, these results can give a sign of the high-frequency performance of the integration algorithm. A previously published Chang family method, named as CWBZI, is considered to compare the overshoot behavior of CVM [30]. As a result, the results for CVM and CWBZ1 for the limiting case of are found to be  
\[\left\{\begin{array}{l}d_{i+1}\approx-\rho d_{i}\\ v_{i+1}\approx\frac{4}{(3+\rho)}v_{i}-\frac{(1-\rho)}{(3+\rho)}v_{i-1}\\ \end{array}\right.\quad\begin{array}{l}CVM\\ \\ \end{array} \tag{21}\] \[\left\{\begin{array}{l}d_{i+1}\approx-\rho d_{i}\\ v_{i+1}\approx-\frac{1}{4}(\rho-1)^{2}\Omega_{0}\omega_{0}d_{i}+\left[\frac{1} {2}(\rho-1)^{2}-1\right]v_{i}\quad CWBZ1.\end{array}\right.\]  
From this equation, we noted that there is no overshoot behavior for any member of CVM in displacement and velocity, and CWBZ1 has no overshoot in the displacement for any value of \(\rho\), while it has a tendency to overshoot the quadratic in \(\Omega_{0}\) in the velocity equation due to the initial displacement term.  
In order to verify the analytical prediction of the overshooting behavior of two methods, the free vibration response of the SDOF system was computed by using an almost large time step. In fact, it was computed by using CVM and CWBZ1 with \(\rho=1\) and \(\rho=0.5\) with a time step corresponding to \(\Delta t/T_{0}\;=\;10\) for the three initial conditions of i. \(d_{0}=1\) and \(v_{0}=0\), ii. \(d_{0}=0\) and \(v_{0}=1\) and iii. \(d_{0}=1\) and \(v_{0}=1\). The numerical findings are displayed in Figures 5-7 for the corresponding initial conditions i., ii. and iii. At the bottom of the Figures 5b, 6b and 7b, the velocity term in the vertical axis is normalized by the natural frequency of the system (\(v_{i}/\omega_{0}\)), in order to have the same unit as displacement. For comparison, the results obtained from the constant average acceleration method (AAM) are also presented in Figures 5-7. It is shown that CVM exhibits no overshoot in both displacement and velocity in all three figures for \(\rho=1\) and \(\rho=0.5\). In the curve for CVM \(\rho=1\) coincides with that of AAM. Similarly, it is displayed on the top plot of Figures 5-7 that the two curves for CWBZ1 exhibit no overshoot in displacement, as the curve for CWBZ1 with \(\rho=1\) coincides with that of AAM. Whereas the bottom part of Figures 5b, 6b and 7b shows a significant overshoot in velocity for CWBZ1 with both \(\rho=1\) and \(\rho=0.5\), although it is almost annihilated in the first few time steps for \(\rho=0.5\). As a result, both analytical and numerical results prove that CVM has no overshoot in displacement and velocity, whereas CWBZ1 has no overshoot in displacement and it has a significant overshoot in velocity. As a summary, these numerical results are in good agreement with the analytical results.  
Figure 5: Comparison of the overshoot response of CVM for the first initial conditions.  
Figure 6: Comparison of the overshoot response of CVM for the second initial conditions.