臺灣博碩士論文加值系統

巨觀車流模式中的波動方程式是將車流行為視為一連續流體所推演出之一雙曲型偏微分方程式。由於雙曲型偏微分方程不易求解，不同的初始條件及不同的邊界條件都將導致不同的解出現，且較複雜的波動方程式其解析解並不存在，需透過數值方法來求解，故本研究探討、比較兩類數值方法：有限差分法以及有限元素法在不同的波動方程模型上之應用，並以此為基礎發展適切的車流波方程有限差分演算法以及有限元素演算法。
本研究利用電子計算機來求解車流波動方程模式之有限差分演算法數值解及有限元素法數值解，並將結果透過電腦繪圖描繪出巨觀車流密度隨著時間、空間的變動情形。本研究比較不同條件下的車流波方程模式之數值解與解析解，研究的結果可以發現，在初始條件不連續的情形下，有限元素法所求得的數值解較有限差分法來得與解析解接近。除此之外，本研究尚利用現有高速公路的流量、速度、密度資料進行探討分析求得車流速度-密度之關係，並將此一關係式帶入車流波方程式中以架構出適合台灣高速公路所使用之車流波方程式，其目的在於檢查兩種不同演算法在實際應用上的可靠度與準確性。
最後，本研究以一簡單的高速公路匝道儀控結果來探討儀控前與儀控後車流密度的變化情形作為車流波方程式在實際上的應用情形。

Macroscopic traffic flow continuum models are compsed of single or systems of partial difference equations (PDEs) with initial and boundary conditions. Due to the difficulty of solving the PDEs problems, numerical methods become a suitable way to find the solution. However, different numerical methods will result in different solutions； how to find an approximate and efficient solution becomes an important course.
This study compares two kind of numerical methods: finite difference method (FDM) and finite element method (FEM) on traffic flow continuum models under different situations. With the aid of computer, this study uses FDM and FEM to solve traffic flow continuum models and plots the variation of traffic density on different time and space.
The result shows that if the initial condition is discontinuous, FEM will have a better result than that of FDM. Besides, this study also compares field data by FDM and FEM and the result also shows that FEM has a better approximation than FDM.
Finally, this study uses a simple example of traffic flow continuum model to explain the before/after ramp metering. This example shows the application of traffic flow continuum models in our real world.

第一章 緒論1
1.1 研究動機1
1.2 研究目的1
1.3 研究範圍2
1.4 研究方法與流程3
第二章 文獻回顧7
2.1 車流波動方程理論之回顧7
2.1.1 一階車流波動方程之回顧7
2.1.2 Newell所發表的累積流量理論9
2.1.3 高階車流波動方程式之回顧9
2.2 車流波方程的分類12
2.2.1 一階波動方程式12
2.2.2 多車道之一階波動方程式14
2.3 車流波方程的數值求解回顧16
第三章 車流模式之推導19
3.1 守恆律之物理觀點19
3.2 車輛數守恆律模式之推導21
3.2.1 名詞定義21
3.2.2簡化型車流守恆率模式之建立22
3.2.3結合車流密度公式的車流模式23
3.3 車流模式之初始條件與邊界條件25
3.3.1 初始條件25
3.3.2 邊界條件27
第四章 車流波方程之求解方法28
4.1 特徵曲線法28
4.2 有限差分法30
4.2.1 有限差分法簡介30
4.2.2 顯式法與隱式法33
4.2.3 計算方法收斂的問題36
4.2.4 常用幾種顯式法之比較38
4.3 有限元素法41
4.3.1 有限元素法簡介41
4.3.2 有限元素法求解44
4.3.3 有限元素法收斂性的探討50
第五章 有限元素法與有限差分法的數值解比較57
5.1 具解析解之一階交通車流模式數值結果比較58
5.1.1 有限差分法的數值結果58
5.1.2 有限元素法的數值結果62
5.2 與實際道路狀況相符之交通車流波方程式之數值解比較64
第六章 道路實際資料的數值解比較72
第七章 高速公路匝道儀控模擬80
第八章 結論與建議84
8.1 結論84
8.2 建議85
參考文獻86

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