In this paper, we deduced a macroscopic traffic model on the uphill and downhill slopes by employing the transformation relation from microscopic variables to macroscopic ones based on a microscopic car-following model considering the velocity difference between adjacent vehicles. The angle [Formula: see text] of the uphill and downhill and the gravitational force have a great impact upon the stability of traffic flow. The linear stability analysis for macroscopic traffic model yielded the stability condition. The Korteweg–de Vries (KdV) equation is derived by nonlinear analysis and the corresponding solution to the density wave near the neutral stability line is obtained. By using the upwind finite difference scheme for simulation, the spatiotemporal evolution patterns of traffic flow on the uphill and downhill are attained. The unstable region is shrunken with slope of the gradient increasing and backward-traveling density waves gradually decrease and even disappear on uphill. Conversely, the unstable region on downhill is extended and density waves propagate quickly backward to the whole road with slope of the gradient increasing.

中文翻译：

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考虑上下坡相邻车辆速度差的宏观交通流模型

在本文中，我们在考虑相邻车辆速度差异的微观跟驰模型的基础上，利用微观变量到宏观变量的转换关系，推导出了上坡和下坡的宏观交通模型。上坡和下坡的角度[公式：见正文]和重力对交通流的稳定性有很大的影响。宏观交通模型的线性稳定性分析得出稳定性条件。通过非线性分析推导出Korteweg-de Vries（KdV）方程，得到中性稳定线附近密度波的相应解。采用迎风有限差分格式进行仿真，得到了上坡和下坡交通流的时空演化规律。不稳定区域随着梯度斜率的增加而缩小，反向传播的密度波逐渐减小，甚至在上坡时消失。反之，下坡不稳定区域扩大，密度波迅速向后传播至整条道路，坡度增大。