On a numerical flux for the pedestrian flow equations*

Abstract

The pedestrian flow equations are formulated as the hyperbolic problem with a source term, completed by the eikonal equation yielding the desired direction of the pedestrian velocity. The operator splitting consisting of successive discretization of the eikonal equation, ordinary differential equation with the right hand side being the source term and the homogeneous hyperbolic system is proposed. The numerical flux of the Vijayasundaram type is proposed for the finite volume solution of the hyperbolic problem. The Vijayasundaram numerical flux, originally proposed for the hyperbolic problems possessing the homogeneity property is extended for pedestrian flow, where the homogeneity property is lost. The application of the proposed numerical flux is demonstrated on the physically relevant problem.

A cellular automaton model for a pedestrian flow problem

“…The curves show the decrease of number of pedestrians in the domain Ω during the evacuation process simulated by CA with various parameters β in the probability matrix (3.2) and by FVM from [19]. The lowest (blue) curve corresponds to the results obtained in [19]. The CA evacuation time T evac corresponding to the various parameters β is shown in the table in Figure 5 (right).…”

“…Here, the FVM evacuation time was T F V M evac = 36.5 s. In Figure 5 (left) the number of pedestrians in the domain Ω against the time t is shown. The curves show the decrease of number of pedestrians in the domain Ω during the evacuation process simulated by CA with various parameters β in the probability matrix (3.2) and by FVM from [19]. The lowest (blue) curve corresponds to the results obtained in [19].…”

“…In Figure 4 the pedestrian distribution is drawn for β = 5 (left) and for β = 20 (right) at time t = 10, 15, 20 s. As can be seen from Figure 4, the evacuation process is faster with an increased value of the parameter β. Since the time of evacuation T evac [s] computed by CA is a significant quantity in the evacuation model, we compare the CA evacuation time with the evacuation time obtained via the finite volume method (FVM) from [19] on the same domain with the number of finite volumes N ELEM = 9056. Here, the FVM evacuation time was T F V M evac = 36.5 s. In Figure 5 (left) the number of pedestrians in the domain Ω against the time t is shown.…”

“…The cost potential is also used in microscopic non-CA models [7,16] and macroscopic models; see e.g. [17,19,27].…”

“…We have been interested in the solution of the pedestrian flow equations from the macroscopic point of view in several papers, namely in [9,19]. We have presented an efficient and robust algorithm for the numerical solution of the room evacuation problem.…”