Efficient numerical methods for multiscale crowd dynamics with emotional contagion
Efficient numerical methods for multiscale crowd dynamics with emotional contagion
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发表时间：
2015
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通讯作者：
Li Wang;M. Short;A. Bertozzi
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作者：
Li Wang;M. Short;A. Bertozzi
本文在具有情绪传染的人群动力学背景下，给出了求解多尺度运动方程的两种有效的数值方法[2]。在连续介质极限中，介观动力学方程产生具有非局域相互作用的自然欧拉极限。然而，当基本的微观粒子特征相交时，这种极限就不再有效，这与欧拉系统中的解的爆炸相对应。一种方法是将这两种情况结合起来--对没有特征交叉的区域使用欧拉动力学，对具有特征交叉的区域使用动力学演化。对于这种混合设置，我们提供了一个基于宏观密度和恐惧程度的制度指标，并提出了一种通过连续性来连接这两个制度的界面条件。另一种方法是基于连续统系统的水平集公式。由此导出的水平集方程与动力学方程具有相似的形式，并且成功地捕获了速度上的多值解，这意味着除了粘性解之外的多值解对于连续介质系统应该是物理上相关的解。数值算例表明了这些新方法的有效性。
In this paper, we develop two efficient numerical methods for a multiscale kinetic equation in the context of crowd dynamics with emotional contagion [2]. In the continuum limit, the mesoscopic kinetic equation produces a natural Eulerian limit with nonlocal interactions. How-ever, such limit ceases to be valid when the underlying microscopic particle characteristics cross, corresponding to the blow up of the solution in the Eulerian system. One method is to couple these two situations – using Eulerian dynamics for regions without characteristic crossing and kinetic evolution for regions with characteristic crossing. For such a hybrid setting, we provide a regime indicator based on the macroscopic density and fear level, and propose an interface condition via continuity to connect these two regimes. The other method is based on a level set formulation for the continuum system. The so-derived level set equation shares similar forms as the kinetic equation, and it successfully captures the multi-valued solution in velocity, which implies that the multi-valued solution other than the viscosity solution should be the physically relevant ones for the continuum system. Numerical examples are presented to show the performance of these new methods.