車流波動方程式數值解法之研究

Abstract

車流波動方程式為一巨觀的車流模型，可以描述道路上的動態車流行 為，以做為道路規劃者的分析工具或即時資訊的提供等。然而，由於車流 波動方程式為一雙曲線型偏微分方程式，其模式之求解十分困難；因此， 要將此一車流波動方程理論有效地應用，發展合理的數值模擬方法實是一 個重要且關鍵的步驟。 本研究針對波動方程的有限差分數值解法作一 完整的探討。首先，針對一階線性波動方程式，以七種有限差分顯式法與 兩種有限差分隱式法進行求解，評比數值解與正確解之間的誤差，而選出 Lax-F、Lax-W及Leapfrog等幾個較佳的演算法進行一階準線性波動方程的 數值模擬。而在準線性波動方程的數值模擬中，我們發現以Lax-F所求得 的解較為合理。 在求解的過程中，我們發現有限差分顯式法必須在滿 足CFL條件的情況下，求得的解才能保證收斂。而為了滿足此一收斂條件 ，必須將空間軸上與時間軸上的切割比增加，亦即必須在時間軸上多做切 割，而造成求解的缺乏效率。因此本研究即以此CFL收斂條件為基礎，發 展一適應性的有限差分法，藉由每一時間層上的密度值來判斷下一個時間 層的切割距離大小，並證明本演算法之收斂性存在。我們並以兩個簡單的 準線性波動方程式為例，分別利用一般的Lax-F有限差分法與本研究所發 展的適應性Lax-F有限差分法進行比較，發現適應性的Lax-F有限差分法只 要以較少的時間軸切割數，便能得到與一般Lax-F差分法同樣精確的結果 。 Macroscopic traffic flow continuum models are composed of single or systems of partial difference equations (PDEs) with suitable initial and boundary conditions which describe various traffic phenomena and road geometry. These models have provided a useful tool with which to test and assess road and traffic control plans. Since the analytical solutions of traffic flow continuum models are difficult to be solved. How to find an approximate and efficient numerical solution becomes an important course. This study takes aim at the numerical finite difference methods of traffic flow continuum models. At first, there are seven methods of explicit finite difference schemes and two methods of implicit schemes used to solve the first order linear continuum models, and compare the errors between exact solution and numerical solutions of these methods. In these results, there are three better algorithms, including Lax-F、Lax-W and Leapfrog schemes, used to simulate the quasilinear continuum models, and the Lax-F scheme can get a more reasonable solution. In the process of numerical computation, we found that every explicit finite difference methods must satisfy the CFL condition to ensure the stability and convergence. This condition requires the ratio of the mesh in space and the mesh in time must satisfy some constrains, and this makes the computation lack of efficiency. Therefore, this study based on the CFL condition develops an adaptive finite difference scheme to solve the LWR model more efficiently. This adaptive scheme can determine automatically the nest suitable time mesh size from the characteristic curve of every grid points in this time, and it can converge to a stable solution. In the numerical test, the Lax method and adaptive Lax method are used to solve the LWR model with different initial and boundary conditions. The simulation results show us that the adaptive Lax method is more efficiency than the Lax method.