Recent advancements in empirical observations of human psychological responses have revealed that the governing rules of human behavioral aspects can not only be predicted with adequate deterministic precision but also forged into mathematical formulations. These modeling studies enable crisis managers with reliable simulation tools to gain insights into critical facets of crowd disasters and enhance their decision-making abilities to facilitate safe egress in emergency situations. The macroscopic scale of representation in this context is often invoked to comprehend large-scale collective pattern formation in vast built environments, owing to its computational viability. The associated governing equations are typically constructed in a system of hyperbolic conservation law form, and conducting simulations in real-life complex geometric configurations can pose computational challenges. This article puts forward a high-resolution, shock-capturing meshfree particle method in an Eulerian framework for the numerical approximation of several widely adopted macroscopic pedestrian flow models. It serves as a superior alternative to traditional mesh-based methods, eliminating the need for specific mesh structures and connectivity between them. A classical Generalized Finite Difference Method (GFDM) in conjunction with several consistency conditions on spatial derivative coefficients is adopted to obtain a non-oscillatory and fairly conservative Godunov-type discretization of the governing system. The intercell flux is evaluated using a suitable Riemann solver, and a slope-limited reconstruction procedure of the corresponding Riemann states, adapted to arbitrary point distribution, is employed for improved accuracy. Moreover, the preferred direction of human motion is determined from a local density-dependent eikonal-type equation, which is solved using a generalized version of the Fast Marching Method. A thorough numerical investigation of three well-known second-order macroscopic models, namely Payne-Whitham (PW), Aw-Rascle (AR), and Aw-Rascle-Zhang (ARZ) is carried out through two hypothetical scenarios of crowd evacuation, not only to demonstrate the accuracy and applicability of the proposed approach but also to deepen understanding of their distinctive characteristics. A comparison of the flow flux with the fundamental diagram suggests that the density profile obtained with the ARZ model is equivalent to Hughes' first-order model. In contrast, the PW and AR models have proven to effectively replicate complex, unstable pedestrian phenomena such as stop-and-go waves. Finally, Helbing's counter-intuitive idea that strategically positioned barriers upstream of a bottleneck can essentially expedite evacuation is numerically investigated across various parameter choices.

中文翻译：

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用于二阶宏观行人流模型数值研究的高分辨率无网格粒子方法

对人类心理反应的实证观察的最新进展表明，人类行为方面的控制规则不仅可以以足够的确定性精度进行预测，而且可以形成数学公式。这些建模研究使危机管理者能够利用可靠的模拟工具深入了解人群灾难的关键方面，并增强他们的决策能力，以促进紧急情况下的安全疏散。由于其计算可行性，这种背景下的宏观表征经常被用来理解巨大建筑环境中大规模集体模式的形成。相关的控制方程通常以双曲守恒定律形式的系统构建，并且在现实生活中复杂的几何配置中进行模拟可能会带来计算挑战。本文在欧拉框架中提出了一种高分辨率、捕捉冲击的无网格粒子方法，用于对几种广泛采用的宏观行人流模型进行数值逼近。它是传统基于网格的方法的优越替代方案，消除了对特定网格结构及其之间连接的需要。采用经典的广义有限差分法（GFDM）结合空间导数系数的几个一​​致性条件，获得了控制系统的非振荡且相当保守的Godunov型离散化。使用合适的黎曼解算器评估单元间通量，并采用适合任意点分布的相应黎曼状态的斜率限制重建程序来提高精度。此外，人体运动的首选方向是根据局部密度相关的 eikonal 型方程确定的，该方程是使用快速行进方法的广义版本求解的。通过两种假设的人群疏散场景，对三个著名的二阶宏观模型，即 Payne-Whitham (PW)、Aw-Rascle (AR) 和 Aw-Rascle-Zhang (ARZ) 进行了彻底的数值研究，不仅证明了所提出方法的准确性和适用性，而且加深了对其独特特征的理解。流量与基本图的比较表明，用 ARZ 模型获得的密度剖面相当于 Hughes 的一阶模型。相比之下，PW 和 AR 模型已被证明可以有效地复制复杂、不稳定的行人现象，例如走走停停的波浪。最后，赫尔宾的反直觉想法是，在瓶颈上游战略性地放置障碍物实际上可以加速疏散，并在各种参数选择上进行了数值研究。