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[Traffic Flow] Concrete example of conservation equation?

  1. Feb 11, 2016 #1
    This is it; most likely the last time I bother the people of this website with my questions on traffic flow.

    I'm trying to figure out some concrete examples to demonstrate utilization of the conservation equation in traffic flow:
    [tex]\frac{\partial \rho }{\partial t} + \frac{\partial q(\rho )}{\partial x} = \frac{\partial \rho }{\partial t} + \frac{\partial \rho v(\rho )}{\partial x} = 0[/tex]
    where ##\rho## is density in ##\frac{num. vehicles}{distance}##, ##q## is flow in ##\frac{num. vehicles}{time}##, ##v## is speed/velocity in ##\frac{distance}{time}##, ##t## is time, and ##x## is distance of a segment of road. ##v(\rho )## can be expressed as follows:
    [tex]v(\rho ) = v_{max}(1 - \frac{\rho }{\rho_{max}})[/tex]
    where ##v_{max}## is maximum velocity and ##\rho_{max}## is maximum density.

    Is there anyone here who knows about traffic modelling well enough to suggest some concrete examples for utilization of this equation? Alternatively, could anyone tell me what variables I would need to know to substitute into the equation in order to get something out?

    I guess my ultimate question here is just, how do I use this equation for modelling traffic now that I have it?
     
  2. jcsd
  3. Feb 14, 2016 #2

    Ssnow

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    The model describe you how the density ##\rho## of cars change in the time ## t##. The fixed data are ##\rho_{max},v_{max}##. After fixing these values the description is given by the equation ##\partial_{t}\rho+\partial_{x}q(\rho)=0##. In order to have a numerical examples you can use the discretization of the equation and approximating derivatives. To find an approximate solution there are a lot of numerical algorithms as Euler, ... (''Numerical Analysis'', Burden & Faires). Interesting can be an analysis of shock waves in this description... I remember you that the model is ##1## dimensional, you consider only one direction in the street that is the ##x##-axis ...
     
  4. Feb 14, 2016 #3

    Svein

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