A deterministic particle approximation for non-linear conservation laws
This work provides a theoretical and numerical framework for approximating conservation laws with particle methods, relevant to applied mathematicians and modelers of traffic/pedestrian dynamics.
The authors developed deterministic particle approximations for non-linear conservation laws, specifically for pedestrian and traffic flow models (LWR and ARZ), using ODE-based schemes. Numerical simulations demonstrate consistency, though no concrete performance numbers are provided.
We review our analytical and numerical results obtained on the microscopic Follow-The-Leader (FTL) many particle approximation of one-dimensional conservation laws. More precisely, we introduce deterministic particle schemes for the Hughes model for pedestrian movements and for two vehicular traffic models, that are the scalar Lighthill-Whitham-Richards model (LWR) and the $2\times2$ system Aw-Rascle-Zhang model (ARZ). Their approximation is performed by a set of ODEs, determining the motion of platoons of possible fractional vehicles or pedestrians seen as particles. Convergence results of the schemes in the many particle limit are stated. The numerical simulations suggest the consistency of the schemes.