On the micro-macro limit in traffic flow
On the micro-macro limit in traffic flow
复制标题
论交通流的微观宏观极限
DOI：
10.4171/rsmup/131-13
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
E. Rossi
中科院分区：
文献类型：
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作者：
R. Colombo;E. Rossi
我们研究了交通流的宏观lighhill - whitham和Richards模型和微观follow-the-leader模型之间的关系。证明了微观模型的解在某种动力学极限下趋向于宏观模型的解，即当个体的总质量不变时，个体的数量趋向于1。基于这一收敛结果，我们利用常微分系统的积分近似地计算了守恒律的解。数学学科分类（2010）。b20 35 l65, 90。关键词。宏观交通模型；微观交通模型；守恒定律。1. 本文的目的是研究交通流微观模型和宏观模型之间的关系。更准确地说，我们考虑lighhill - whitham[7]和Richards[10]连续体模型@tr @x r v(r) 0 r(0; x) ~r(x)(1:1)，其中t2r为时间，x2r为空间坐标，r r（t; x）为（平均）车辆密度，其中r 2 [0；1], v2c （[0; 1]; [0; v]）为宏观速度定律。在离散级，我们选择一阶（*）Indirizzo dell'A。：布雷西亚大学INdAM单元，braze38,25123布雷西亚，意大利。E-mail: Rinaldo.Colombo@UniBs.it (**) Indirizzo dell'A。：米兰-比可卡大学数学与应用系，via Cozzi 55, 20125 Milano, Italy。E-mail: e.rossi50@campus.unimib.it Follow the Leader [1] model _ pi w(pi1 * pi) i1；……；n _ pn1 V (0) ~ i 1；……；N 1 ><>: (1:2) where p1；……;pn1为n1个驾驶员的位置，其中pi1½pi ‘, ’为车辆的平均长度，w2c （';1 [; [0;V]）为微观速度规律。请注意，需要指定最前面车辆的速度，但这个速度不一定是最大速度。宏观描述（1.1）和微观描述（1.2）是通过粒子路径联系起来的，粒子路径是单个个体根据（1.1）的轨迹，即常微分方程_ p v r(t; p(t)) p(0) ~ p的解；（1:3），因为~ p随R的变化，（1.3）的完备性见[4]。宏观模型（1.2）和微观模型（1.1）之间的联系在于，（1.2）中的每个方程都是（1.1）的粒子路径（1.3）。唯一的例外是最前面的车辆，它的速度是指定的。下面，通过特设算子，我们建立了宏观变量r与微观变量（p1；…）之间的关系。;Pn1)，表明这两种描述在某种程度上是镜面的。然后，我们证明了（1.1）的解在某种动力学极限下趋向于（1.2）的解，即当个体数趋于1而n '保持不变时。从建模的角度来看，这一结果证明了LighthillWhitham and Richards模型（1.1）是一阶追随领导者模型的极限，因为个体数量趋向于1。对于强调汉密尔顿-雅可比方程的相关方法，我们参考[5]。此外，上面概述的极限程序建议将常微分方程（1.2）也用作偏微分方程（1.1）的数值积分工具，参见第3节。下面将彻底研究这种可能性。几个集成说明了严格的结果。然而，基于常微分系统（1.2）的数值解来计算（1.1）解的数值算法很难与特别方法（如经典的Lax-Friedrichs方法）竞争。下一节介绍主要的分析结果。第3节专门讨论（1.1）和（1.2）的几个数值积分。所有技术细节将推迟到最后的第4节。218 R.M.科伦坡E.罗西
We investigate the relations between a macroscopic Lighthill-Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. Solutions to the microscopic model are proved to tend to those to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to1 while their total mass is constant. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system. MATHEMATICS SUBJECT CLASSIFICATION (2010). 35L65, 90B20. KEYWORDS. Macroscopic traffic models; microscopic traffic models; conservation laws. 1. Introduction Aim of this paper is to investigate the relations between a microscopic and a macroscopic model for traffic flow. More precisely, we consider the Lighthill-Whitham [7] and Richards [10] continuum model @tr @x r v(r) 0 r(0; x) ~r(x) ( (1:1) where t 2 R is time, x 2 R is the space coordinate, r r(t; x) is the (average) vehicular density, with r 2 [0; 1], and v 2 C([0; 1]; [0;V ]) is the macroscopic speed law. At the discrete level, we choose the first-order (*) Indirizzo dell'A.: INdAM Unit, University of Brescia, via Branze 38, 25123 Brescia, Italy. E-mail: Rinaldo.Colombo@UniBs.it (**) Indirizzo dell'A.: Department of Mathematics and Applications, University of Milano-Bicocca, via Cozzi 55, 20125 Milano, Italy. E-mail: e.rossi50@campus.unimib.it Follow the Leader [1] model _ pi w(pi1 ÿ pi) i 1; . . . ;n _ pn1 V pi(0) ~ pi i 1; . . . ;n 1 ><>: (1:2) where p1; . . . ; pn1 are the positions of the n 1 drivers, with pi1 ÿ pi `, ` being the average vehicle's length, and w 2 C( `;1 [; [0;V ]) is the microscopic speed law. Remark that it is required to assign the speed of the foremost vehicle, but it is not necessary that this speed be the maximal one. The macroscopic description (1.1) and the microscopic one (1.2) are related through particle paths, which are the trajectories of single individuals according to (1.1), namely the solutions to the ordinary differential equation _ p v r(t; p(t)) p(0) ~ p ; ( (1:3) as ~ p varies in R, see [4] for the well posedness of (1.3). The connection between the macroscopic model (1.2) and the microscopic one (1.1) consists in imposing that each equation in (1.2) is the particle path (1.3) for (1.1). The only exception is the foremost vehicle, whose speed is assigned. Below, through ad hoc operators, we establish a relation between the macroscopic variable r and the microscopic one (p1; . . . ; pn1), showing that the two descriptions are to some extent specular. Then, we show that solutions to (1.1) tend to solutions to (1.2) in a sort of kinetic limit, i.e. as the number of individuals tends to 1 while n` remains constant. From the modelling point of view, this result justifies the LighthillWhitham and Richards model (1.1) as the limit of a first order follow the leader model, as the number of individuals tends to 1. For a related approach with emphasis on the Hamilton-Jacobi equation we refer to [5]. Besides, the limiting procedure outlined above suggests the use of the ordinary differential equations (1.2) also as a tool for the numerical integration of the partial differential equation (1.1), see Section 3. This possibility is thoroughly investigated below. Several integrations illustrate the rigorous results. However, a numerical algorithm to compute the solutions to (1.1) based on the numerical solution to the ordinary differential system (1.2) hardly competes with an ad hoc method, such as the classical Lax-Friedrichs method. The next section presents the main analytical results. Section 3 is devoted to several numerical integrations of both (1.1) and (1.2). All technical details are deferred to the final Section 4. 218 R.M. Colombo E. Rossi