The theoretical analysis of the anticipation lattice models for traffic flow

Abstract

Based on the anticipation lattice hydrodynamic models, which are described by the partial differential equations, the continuum version of the model is investigated through a reductive perturbation method. The linear stability theory is used to discuss the stability of the continuum model. The Korteweg–de Vries (for short, KdV) equation near the neutral stability line and the modified Korteweg–de Vries (for short, mKdV) equation near the critical point are obtained by using the nonlinear analysis method. And the corresponding solutions for the traffic density waves are derived, respectively. The results display that the anticipation factor has an important influence on traffic flow. From the simulation, it is shown that the traffic jam is suppressed efficiently with taking into account the anticipation effect, and the analytical result is consonant with the simulation one.

Appendix

Substituting Eqs. (12)–(13) into Eq. (10), then we get the linearized equation,

$$\begin{aligned} &{\partial^{2}_{t} y+a\partial_{t} y+a \rho_{0}^{2}V'(\rho _{0})y(x+1)} \\ &{\quad+sa \rho_{0}^{2}V'(\rho_{0}) \bigl(y(x+1)-y(x)\bigr)=0,} \end{aligned}$$

(38)

where V′(ρ ₀)=dV(ρ)/dρ, at ρ=ρ ₀. Expanding y(x,t)=exp(ikx+zt), we have

$$\begin{aligned} &{z^{2}+az+a\rho_{0}^{2}V' \exp(ik)} \\ &{\quad+iksa\rho_{0}^{2}V'\bigl(\operatorname{exp}(ik)-1 \bigr)=0.} \end{aligned}$$

(39)

Prescribing z=z ₁(ik)+z ₂(ik)²+⋯ and $\exp(ik)=1+ik+\frac{1}{2}(ik)^{2}+\cdots$, inserting them into Eq. (38), and keeping the first- and second-order terms of ik, then we get the coefficients of z, respectively,

$$\begin{aligned} &{z_{1}=-\rho_{0}^{2}V',\qquad z_{2}=-\tau\bigl(\rho_{0}^{2}V' \bigr)^{2}-(s+1)\rho_{0}^{2}V'.} \end{aligned}$$

(40)

If z ₂ is a positive value, the uniform flow is stable. If z ₂ is a negative value, the uniform flow is unstable for long wavelength modes. So we get the unstable condition

$$\begin{aligned} a<\frac{-\rho_{0}^{2}V'}{1+s}. \end{aligned}$$

(41)