In this paper we propose a novel macroscopic (fluid dynamics) model for describing pedestrian flow in low and high density regimes. The model is characterized by the fact that the maximal density reachable by the crowd - usually a fixed model parameter - is instead a state variable. To do that, the model couples a conservation law, devised as usual for tracking the evolution of the crowd density, with a Burgers-like PDE with a nonlocal term describing the evolution of the maximal density. The variable maximal density is used here to describe the effects of the psychological/physical pushing forces which are observed in crowds during competitive or emergency situations. Specific attention is also dedicated to the fundamental diagram, i.e., the function which expresses the relationship between crowd density and flux. Although the model needs a well defined fundamental diagram as known input parameter, it is not evident a priori which relationship between density and flux will be actually observed, due to the time-varying maximal density. An a posteriori analysis shows that the observed fundamental diagram has an elongated "tail" in the congested region, thus resulting similar to the concave/concave fundamental diagram with a "double hump" observed in real crowds. The main features of the model are investigated through 1D and 2D numerical simulations. The numerical code for the 1D simulation is freely available at https://gitlab.com/cristiani77/code_arxiv_2406.14649

翻译：本文提出了一种新颖的宏观（流体动力学）模型，用于描述低密度与高密度状态下的行人流。该模型的特点是人群可达到的最大密度——通常是一个固定的模型参数——在此作为一个状态变量。为实现这一点，该模型将用于追踪人群密度演化的守恒定律（按常规方式构建）与一个描述最大密度演化的、带有非局部项的类Burgers偏微分方程相耦合。这里使用可变最大密度来描述在竞争或紧急情况下人群中观察到的心理/物理推挤力的影响。研究还特别关注基本图，即表达人群密度与流量之间关系的函数。尽管模型需要一个明确定义的基本图作为已知输入参数，但由于最大密度随时间变化，密度与流量之间实际将呈现何种关系并非先验可知。后验分析表明，观测到的基本图在拥堵区域具有一个延长的“尾部”，从而使其类似于在实际人群中观察到的具有“双峰”的凹/凹型基本图。通过一维和二维数值模拟研究了模型的主要特征。一维模拟的数值代码可在 https://gitlab.com/cristiani77/code_arxiv_2406.14649 免费获取。