The two-dimensional Godunov scheme and what it means for continuum pedestrian flow models

Abstract

we show the results of three of them. In the full paper, there will be more examples and the results will be discussed more in-depth. In the test cases, the pedestrians are initially located in a circular area. As Figure 1 shows, they are then attracted to the centre (case 1 and 3) or they are repelled from the centre (case 2). In case 3, there is furthermore a second class that moves from left to right, through the area with the pedestrians of the first class.

(a) Case 1 (b) Case 2 (c) Case 3. Dashed arrows: class 1, solid arrows: class 2.

Figure 1: Initial positioning (gray areas) and walking direction (arrows).

The test results (Figure 2) show that the numerical method can deal well with flows where densities strongly increase (e.g. towards the centre in case 1), where densities strongly decrease (e.g. away from the centre in case 2) and with flows of different directions (case 3). Our preliminary results for case 1 and 2 show almost perfect rotational symmetry, indicating only little influence of the walking direction and grid orientation on the computed velocity. Furthermore, preliminary results for case 3 (not shown) indicate an influence of the resolution (grid cell size) on the width of the lanes that emerge. This shows that numerical resolution has to be considered in relation to the scale at which one desires to reproduce phenomena: small scale phenomena can not be accurately reproduced on a coarse grid. The details of this finding are discussed in the full paper.

(a) Case 1. (b) Case 2.

(c) Case 3, class 1 (towards centre). (d) Case 3, class 2 (left to right).

Figure 2: Space density plot at time t = 20 s.

3 Outlook for full paper In the full paper, we discuss the simulation method, the tests and their results in more detail. We show that self-organising phenomena may occur differently with different numerical settings, indicating that a good setting is important. Two aspects are studied in detail. Firstly, the orientation of the grid may have an influence on the occurrence of self-organisation. Secondly, the size of the grid cells may have an influence on the scale at which self-organisation occurs, e.g. the width of emerging lanes. Future research includes more detailed comparison of the proposed numerical method with other methods. Furthermore, the two-dimensional Godunov method will and already has been applied in (future) research to study continuum pedestrian flow models [2, 3]. Since one of the objectives of these studies is to investigate selforganisation, it is important to apply an accurate numerical method such as the one proposed in this contribution.

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