動態車流方程式數值解之研究－以LWR及其包含擴散項之模式為例

Abstract

動態的車流行為可以做為道路規劃者有力的分析工具，或是當成即時資訊的提供來源。然而，車流波動方程式為一雙曲型偏微分方程式，求解不易，不同的初始和邊界條件都會導致不同的結果出現，較複雜的波動方程式甚至於不存在解析解。所以，要廣泛的應用車流波動方程，必須透過數值模擬方法來求解。 有鑑於以往求解車流波動方程，大部分均利用一階準確的有限差分數值方法求解，使得準確度並不高。本研究乃針對有限差分方法，嘗試利用準確度較高且數值結果不發生震動的TVD數值方法，及近幾年來才發展的高階準確ENO數值方法，來求解車流問題。並比較於初始條件為不連續的情況下，利用TVD、ENO方法及傳統一階準確Lax-F法、Godunov方法和二階準確的Lax-W法，求解具解析解的準線性車流問題。結果發現TVD和ENO數值方法明顯的優於其它的方法。此外，並考慮包含擴散項的LWR延伸車流模式，利用數值方法模擬求解，分析其交通意義。 最後，考慮道路為具有車輛進出交通狀況的非線性車流問題，而此車輛進出入項為密度、空間和時間的函數，以數值方法模擬求解此複雜的非線性車流問題，比較此進出入項對於結果的影響，並解釋其於交通上的行為。

Macroscopic traffic flow continuum models are partial difference equations (PDEs) with initial and boundary conditions. Since the analytical solutions of traffic flow continuum models are difficult to be solved, 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. In the past, solving numerical solution of traffic flow continuum models is usually used first order approximation numerical methods, giving lower accurate. This study takes aim at numerical finite difference methods. Use numerical methods that are at least second order accurate on smooth solution and yet give well resolved , nonoscillatory discontinuities – TVD method, and uniform high order accurate methods – ENO methods, to solve traffic flow continuum models problem. In these result, we find that TVD and ENO numerical is obviously better than traditional methods. In addition, we consider LWR with diffusion term model problem, and using numerical methods to solve it. Finally, this study will consider nonlinear problem. It adds a reaction term to LWR Model. This study use numerical methods to solve these complex nonlinear problem.