平行模擬車流波茲曼方程式之研究

Abstract

動態車流的研究自從Lighthill 與 Whitham在五十年代提出了以波動方程式來描述交通車流行為的模式，愈來愈多學者投入用數學方程式來描述交通車流行為的研究。且根據不同之交通現象與道路狀況所建立的相關模式與研究也被陸續提出。 動態車流模式可概略分為三大類，巨觀、微觀和本研究所欲探討的中觀車流模式。最先提出中觀車流模式的是Prigogine和Andrews兩位學者。他們修改了一些在氣體動力學中的概念，導出一描述車流行為的似波茲曼方程式。本研究將提出一個新的交通波茲曼方程式，利用速度分佈函數來描述動態車流。在此交通波茲曼方程式，我們考慮車子受漂移、交通場、加減速和換車道所控制。由於波茲曼方程式是個高度複雜的方程式，本研究將使用蒙地卡羅模擬法來求解。 蒙地卡羅模擬法是利用隨機的方法來模擬一個系統。在本研究中，蒙地卡羅模擬法乃是利用電腦對車子的運動做直接的物理模擬(direct physical simulation)，而非去解車流運動數學模型的數值解。雖然蒙地卡羅模擬法較精確的描述車子運動行為，但其在模擬的效率上來說並不是很好。所以最後我們就將利用個人電腦叢集計算的技術來改善蒙地卡羅模擬效率。

Lighthill and Whitham proposed their kinematic traffic flow model five decades ago, then the mathematical description of traffic flow operations has been a lively subject of research and debate for traffic scientists. There were a wide range of traffic flow theories and models, which were developed to describe traffic flow operations. In this study, we focus on mesoscopic traffic flow model. The gas-kinetic traffic flow model of Prigogine and Andrews, one of mesoscopic traffic flow model, was developed during 1960’s. They modified some of the key concepts in the kinetic theory of gases and wrote down an equation alike to the Boltzmann transport equation. We present a new traffic Boltzmann transport equation describes the dynamics of the velocity distribution functions of vehicles in the traffic flow. From the traffic Boltzmann transport equation, we consider the vehicles governed by drift, traffic field, deceleration, and lane-changing. The Monte Carlo simulation technique plays an important role in solving the complex equation. The name of Monte Carlo simulation technique is usually given to stochastic methods that employ a stochastic process to simulate a system. In this study, Monte Carlo simulation technique is introduced to directly solve the traffic Boltzmann transport equation by direct physical simulation. Monte Carlo simulation technique offers an accurate description of transport, but it requires intensive computation and hence has not found wide use for traffic flow applications. Then we introduce the parallel Monte Carlo simulation technique used for improving the drawback of Monte Carlo simulation. Finally, comparison the simulation results with real VD data is discussed.