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Implementation of the vehicular occupancy-emission relation using a cubic B-splines collocation method

  • * Corresponding author: Sofiya Chergui

    * Corresponding author: Sofiya Chergui 
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  • The complexity and non-linearity of flow phenomena are explained by numerous criteria, including the interactions of the large number of vehicles occupying the road, which influence the road density. This density under certain conditions, leads to traffic congestion which has dangerous effects on the environment such as; resources consumption; noise and the effect caused by greenhouse gas emissions of the $ CO_{2} $ and other pollutants. In this paper we consider working in an uniform, homogeneous road where the traffic is described by the Lighthill Whitham-Richard (LWR) model resolved using a cubic B-spline collocation scheme in space and an implicit Runge Kutta scheme in time. We also shed light on the relation between vehicle occupancy and vehicle emissions.

    Mathematics Subject Classification: 58J45, 35L02; Secondary: 35L65, 35L60.


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  • Figure 1.  Approximate density (veh/m)

    Figure 2.  Approximate density. Vs exact density (veh/m)

    Figure 3.  The variations of Link occupancy in time

    Figure 4.  The evolution of hydrocarbon emission rate in time

    Figure 5.  Aggregate emission rate vs. link occupancy

    Figure 6.  Flow.Vs density (fundamental diagram)

    Figure 7.  Travel speed

    Table 1.  coefficients of $ B_{j} $ and its derivatives

    $ x $ $ x_{j-1} $ $ x_{j} $ $ x_{j+1} $
    $ B_{j} $ $ \frac{1}{6} $ $ \frac{4}{6} $ $ \frac{1}{6} $
    $ B^{'}_{j} $ $ \frac{-1}{2h} $ $ 0 $ $ \frac{1}{2h} $
    $ B^{''}_{j} $ $ \frac{1}{h^{2}} $ $ \frac{-2}{h^{2}} $ $ \frac{1}{h^{2}} $
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