We study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation $\parallel \nabla \phi \parallel =f$, with a positive continuous function $f$ connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock’s primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.

我们研究了人群运动数学预测校正模型的新变体。预测阶段由传输方程处理，其中矢量场通过 eikonal 方程计算$\parallel \nabla \phi \parallel =F$，具有正连续函数$F$与自发旅行的速度有关。修正阶段由新版本的最小流量问题处理。该模型非常灵活，可以考虑代理之间不同类型的相互作用，从 Wassersetin 空间中的梯度流到沙堆中的颗粒类型动力学。此外，可以使用不同的边界条件，例如非齐次狄利克雷（例如具有不同退出成本惩罚的外出）和诺伊曼边界条件（例如具有不同速率的入口）。结合输运方程的有限体积法和用于演算方程和最小流量问题的 Chambolle-Pock 原始对偶算法，我们提出了数值模拟来演示不同场景下的行为。