## 4 Numerical Simulation and Discussion  
Here, we have conducted numerical simulations to verify the properties of the DAVD model presented in Equation (5). Situation 1:  
The initial simulation environment is built as follows: overall vehicle number is \(N=50\), which are distributed on a single lane ring road with length \(L=1000\) m. The initial position of car \(n\) is settled as:  
\[\left\{ \begin{array}{cc}x_{n}(0)=1,&n=1\\ x_{n}(0)=(n-1)L/N,&n=2,3,\ldots,N\end{array} \right., \tag{17}\]  
According to Equation (17), the first car's position is 1 m instead of 0 m, and the other vehicles are evenly distributed at the same distance of 20 m. Therefore, the initial small disturbance is set up as the initial gap between the cars 50 and 1 is 21 m, and the initial gap between the cars 1 and 2 is 19 m. Based on the velocity optimal function as in Equation (6), the initial velocity of car \(n\) is \(v_{n}(0)=V(L/N)\). Update rules for position and velocity are:  
\[\left\{ \begin{array}{cc}v_{n}(t+\Delta t)=v_{n}(t)+\Delta t\cdot\frac{d^{2} x_{n}(t)}{dt^{2}}\\ x_{n}(t+\Delta t)=x_{n}(t)+\Delta t\cdot\frac{v_{n}(t)+v_{n}(t+\Delta t)}{2} \end{array} \right., \tag{18}\]  
To compare with the FVD model, the relevant parameters of this model in the simulation are consistent with those in the literature [9], \(\alpha=0.41\), \(\lambda=0.5\), time step \(\Delta t=0.1\) s.  
The DAVD model experiments are performed under the different parameters affecting the headway evolution. Figure 3 shows traffic density waves at time intervals of \(t=0\)-500 s and t = 1800-2000 s. In Figure 3a,b, the DAVD model is transformed into the FVD model corresponding to the case of \(\beta=0\), \(p=0\), \(m=1\). It is obviously found that the initial disturbances adding into the uniform traffic flow propagate backwards faster and faster, and meanwhile the fluctuation range of the headway is becoming larger. Eventually, the uniform flow develops into a chaotic traffic flow. When the communication is interrupted in the real world, the influence coefficient of traffic density in the DAVD model decreases, and only the state of the preceding vehicle can be observed. In this case, the parameters of the DAVD model are reset to \(\beta=0.1\), \(p=0.1\), \(m=1\). Comparing to Figure 3a,b, the traffic congestion is well alleviated in Figure 3c,d. Before the time steps of \(t=500\)s, the vibration magnitude of the headway is inapparent, which means the propagation speed of the perturbations has decreased in this case. After \(t=1800\)s, although the waves of stop-and-go in traffic flow gradually appear and continue to propagate, the density waves amplitude is decreased. Figure 3e,f indicate that the oscillation amplitude of the headway is significantly reduced at different time intervals. In addition, density waves almost disappear with the increase of the parameters \(\beta\), \(p\), \(m\). Therefore, Figure 3 demonstrates that under the influence of external disturbances, the traffic flow in the model will quickly return to a methodic and stable state with increasing attention to acceleration and traffic density information. To further prove the behavior of the DAVD model in different cases, the standard deviations of the velocity distributions are obtained to reflect the discreteness of the vehicle speed distribution and measure traffic flow stability at various values of parameters. Figures 4 and 5 show the variation of the velocity standard deviation and the standard deviation of each car's velocity under three different parameters, respectively. In Figure 4, the standard deviation curves of patterns (a) and (b) can be divided into three stages, which are slowly rising, rising rapidly and tending to stabilize. In contrast, the curve of pattern (c) is similar to a straight line and the value of standard deviation is close to 0. From the perspective of each car, the fluctuation range of each car's velocity of patterns (a) and (b) is larger than pattern (c).