A kinetic theory approach to model pedestrian dynamics in bounded domains with obstacles

Abstract

We consider a kinetic theory approach to model the evacuation of a crowd from bounded domains. The interactions of a person with other pedestrians and the environment, which includes walls, exits, and obstacles, are modeled by using tools of game theory and are transferred to the crowd dynamics. The model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously in a panic situation. For the numerical approximation of the solution to our model, we apply an operator splitting scheme which breaks the problem into two pure advection problems and a problem involving the interactions. We compare our numerical results against the data reported in a recent empirical study on evacuation from a room with two exits. For medium and medium-to-large groups of people we achieve good agreement between the computed average people density and flow rate and the respective measured quantities. Through a series of numerical tests we also show that our approach is capable of handling evacuation from a room with one or more exits with variable size, with and without obstacles, and can reproduce lane formation in bidirectional flow in a corridor. arXiv:1901.07620v3 [math.NA] 14 Jun 2019 density and linear momentum, which are regarded as macroscopic observables of pedestrian flow. See, e.g., [28,37]. Such an approach is suitable for high density, large-scale systems, which are not the focus of our work.A second approach looks at the problem at the microscopic level. Microscopic models can be further divided into models which are grid-based or grid-free. Cellular Automata [14,15,16,21,31] models belong to the first category. They describe pedestrian flow in space-time by assigning discrete states to a grid of space-cells. These cells can be occupied by a pedestrian or be empty. Thus, the movement of pedestrians in space is done by passing them from cell to cell (discrete space) in discrete time. Grid-free methods can be based on second order models (forces-based), first order models (vision-based or speed-based) or zeroth order models (rule-based or decision-based). Forcebased models use Newtonian mechanics to interpret pedestrian movement as the physical interaction between the people and the environment, i.e. the action of other people and the environment on a given pedestrian is modeled with forces. These models are one of the most popular modeling paradigms of continuous models because they describe the movement of pedestrians qualitatively well. See, e.g., [18,24,26,27,32,33,39,43] and references therein. Collective phenomena, like unidirectional or bidirectional flow in a corridor, lane formation, oscillations at bottlenecks, the faster-is-slower effect, and emergency evacuation from buildings, are well reproduced. Agentbased models allow for flexibility, extensibility, and capability to realize heterogeneity in crowd dynamics. For examples of vision-based, speed-based, rule-based, and decision-based models we refer to [2,3, 4,17,19,20] and references t...

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