連續流車流模式的有限差分近似解法

ON THE APPROXIMATION OF FINITE DIFFERENCE METHODS FOR CONTINUUM TRAFFIC FLOW MODELS

連續流車流模式屬雙曲線型偏微分方程式，求解不易，因此，一般多藉由有限差分法加以近似求解。本研究以均勻到達車流碰上停等車隊產生向上游回溯衝擊波，以及停等車隊起動向下游疏解等兩個嚴苛的交通狀況為例，進行多種有限差分方法在不同階等連續流模式的適用性評比。結果發現，Lax-F有限差分法無論是針對一階準線性連續流模式（即LWR模式）或高階連續流模式，均能獲致與解析解最為相近的近似解，而且安定性與收斂性亦佳，適用性最佳。此與以往國內外研究略有不同，主要係因以往研究所研析的交通案例均侷限於非壅塞交通狀態及Payne模式本身缺陷所致。

Because continuum traffic flow models are described by hyperbolic partial differential equations and the exact solutions to the models are difficult to derive analytically, finite difference methods are commonly used to approximate the solutions. In this paper, the fitness of several finite difference methods under different continuum flow models is assessed in two extreme traffic conditions, a backward shock wave formed by a uniform arrival flow encountering a stopped flow and a forward shock wave formed by a discharging flow. The results show that the Lax-F method can obtain good approximations for both the first-order quasi-linear continuum model (i.e., the LWR model) and the high-order continuum flow model with good stability and convergence. This finding is different from previous domestic and foreign studies primarily because past studies examined only traffic cases with uncongested traffic conditions and because of the drawbacks of the Payne model itself.