Journal
MATHEMATICAL MODELLING OF NATURAL PHENOMENA
Volume 17, Issue -, Pages -Publisher
EDP SCIENCES S A
DOI: 10.1051/mmnp/2022023
Keywords
Specified homogenization; Hamilton-Jacobi equations; pedestrians; non-local operators; Slepcev formulation; viscosity solutions; microscopic model on junction
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Funding
- European Union
- European regional development fund (ERDF) [18P03390/18E01750/18P02733]
- Normandie Regional Council via the M2SiNUM project
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This paper establishes a rigorous connection between microscopic and macroscopic pedestrian models at a convergent junction. By injecting the microscopic model into a non-local PDE, the study demonstrates the local uniform convergence of the viscosity solution of the non-local PDE towards the solution of the macroscopic model.
In this paper, we establish a rigorous connection between a microscopic and a macroscopic pedestrians model on a convergent junction. At the microscopic level, we consider a follow the leader model far from the junction point and we assume that a rule to enter the junction point is imposed. At the macroscopic level, we obtain the Hamilton-Jacobi equation with a flux limiter condition at x = 0 introduced in Imbert and Monneau [Ann. Sci. l'ecole Normale Super. 50 (2017) 357-414], To obtain our result, we inject using the cumulative distribution functions the microscopic model into a non-local PDE. Then, we show that the viscosity solution of the non-local PDE converges locally uniformly towards the solution of the macroscopic one.
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