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# Macroscopic Model

Macroscopic models represent pedestrains as a whole, individual pedestrains are neglected. These models give information about the Pedestrainflow and the densities.

The macroscopic models are described by partial differential equations. There are different ways one can follow. In the following we will present these different models.

## Euler-/Navier-Stokes Equations

One idea is to use Euler-/Navier-Stokes differential equations, which are known from the field of fluid dynamics. The motivation to use these equations for the modeling of pedestrian flows follows from the phenomenological similarity of flowing fluids to "flowing" pedestrian crowds. An important characteristic here is the conservation of mass, which is automatically satisfied by the equations of fluid dynamics. In addition, there are many well-studied solution methods for these models.

The use of the above mentioned fluid equations however, also involve drawbacks. The equations consist of a transport equation to describe the conservation of mass and an equation that ensures the conservation of momentum. The behavior of people is, however, not motivated by a physical momentum to conserve, as for instance in case of non independent operating particles in fluids. One can now try to adjust the momentum equation so that plausible human behavior is approximated.

[Bärwolff, G., Chen, M.-J., Schwandt, H., Slawig, T.: Simulation of pedestrian flow for traffic control systems]

## Multiphase Flow

The above partial differential equations of fluid dynamics are single-phase-models, that is, the velocities of all persons modeled in a local area are considered as equal. This limits, however, the modeling of different behavior of specific groups of individuals. For this purpose our project group has modified a multi-phase solver, so that it is thus possible to simulate several different groups of pedestrians.

[Huth, F. and Bärwolff, G. and Schwandt, H.: An extended multi-phase transport model for pedestrian flow]

## Custom Model

Besides the use of already known models, such as that of fluid dynamics, it is also possible to construct a completely new model. This was done by omitting the above mentioned momentum equation and extending the transport equation by further equations to describe human behavior. For example, a flow term was introduced, which depends on the density gradient. This ensures that pedestrians within large collections tends to spread, which is an important aspect of human behavior. The direction of this movement is calculated temporally dynamic. This is done by a potential field which contains information about the environment. This information determines the decision of the pedestrians for a new meaningful direction. As an example, remote information such as information about the geometry of the area or pedestrian traffic jams slip in the information field. In addition local information such as not jammed other pedestrians can slip in.

This model is an n-species model, thus it is also possible to simulate the behavior of different groups of people. The model was extended by a term which describes the formation of groups, which means that people which are from the same group are more cohesive.