Traffic equation

The traffic equation is a partial differential equation , more precisely a non-linear hyperbolic wave equation , with which traffic models can be simulated. Road traffic, which can be simulated with the traffic equation, can be considered as an illustrative example. It relates the change in traffic flow over time to the change in density along the route.

The subscripts represent the derivation of the variable according to a certain variable, for example the derivation of the density according to the position variable. Let the density of a stream at the location at the time and the corresponding flow, which usually depends on the traffic density and the speed of the elements, then the traffic equation reads: ${\ displaystyle \ varrho _ {x} (x, t)}$ ${\ displaystyle \ varrho (x, t)}$ ${\ displaystyle x}$ ${\ displaystyle t}$ ${\ displaystyle f (x, t)}$

{\ displaystyle \ varrho _ {t} (x, t) + f _ {\ varrho} (\ varrho (x, t)) \ cdot \ varrho _ {x} (x, t) = 0 \ qquad \ qquad \ forall x, t}

In the simplest case, the flow is a linear combination of speed and density. The more particles there are on the way and the faster they move, the higher the throughput.

{\ displaystyle f = \ varrho \ cdot v}

However, this formula cannot be used for road traffic, as cars keep a greater distance from one another at higher speeds and the density therefore decreases. If the density decreases with increasing speed, the result is:

{\ displaystyle f (\ varrho) = v_ {max} \ cdot \ varrho \ left (1 - {\ frac {\ varrho} {\ varrho _ {max}}} \ right)}

The river forms a parabola that reaches its maximum at half its maximum speed. For all other values there are two possibilities in this version to achieve them: through high density at low speed or vice versa. The aim is usually to achieve the second variant, since a high average speed keeps the length of time the particles stay short.

An important variable here is the derivation of the flow according to the density, which is called the signal speed . ${\ displaystyle f _ {\ varrho}}$

Derivation

The traffic equation is based on the conservation law of traffic elements . Their total number can be determined from the density along the entire route:

{\ displaystyle n (t) = \ int _ {a} ^ {b} \ varrho (x, t) dx}

The change in the number of elements over time can also be tracked. The particles must enter the considered path at the starting point or leave at the end point , which can be found out using a balance calculation: ${\ displaystyle n_ {t} (t)}$ ${\ displaystyle a}$ ${\ displaystyle b}$

{\ displaystyle \ int _ {a} ^ {b} \ varrho _ {t} (x, t) dx = f (a, t) -f (b, t)}

The change in density over time must therefore be compensated for by a change in the flow in the place, and that at any point in time.

{\ displaystyle \ int _ {a} ^ {b} \ left (\ varrho _ {t} (x, t) + f_ {x} (x, t) \ right) dx = 0}

This not only has to apply at all times, but also for any route. If one assumes sufficient differentiability, the equation can be represented in an integral-free representation:

{\ displaystyle \ varrho _ {t} (x, t) + f_ {x} (x, t) = 0 \ qquad \ qquad \ forall x, t}

Since the traffic flow f can be represented as a function of the density, the following differentiation results:

{\ displaystyle \ varrho _ {t} (x, t) + f _ {\ varrho} (\ varrho (x, t)) \ cdot \ varrho _ {x} (x, t) = 0 \ qquad \ qquad \ forall x, t}

literature

Hans-Joachim Bungartz, Stefan Zimmer, Martin Buchholz, Dirk Pflüger: Modeling and simulation. An application-oriented introduction. Springer-Verlag, Berlin et al. 2009, ISBN 978-3-540-79809-5 ( eXamen.press ).