6533b832fe1ef96bd129af21

RESEARCH PRODUCT

Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux

Carlotta DonadelloUlrich RazafisonMassimiliano D. RosiniBoris Andreianov

subject

Crowd dynamicsMathematical optimizationFixed point argumentsDiscretizationGeneral MathematicsScalar (mathematics)Crowd dynamics; Finite volume approximation; Nonlocal point constraint; Scalar conservation law; Vehicular traffics; Well-posedness; Mathematics (all); Applied Mathematics01 natural sciencesMSC : 35L65 90B20 65M12 76M12NONonlocal point constraintCrowdsData acquisitionMathematics (all)[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]DoorsUniqueness[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematicsScalar conservation lawMathematicsConservation lawVehicular trafficsFinite volume methodApplied Mathematics010102 general mathematics[MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA]010101 applied mathematicsWell-posednessFinite volume schemeFinite volume approximationConvergence of approximations[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]

description

We introduce and analyze a class of models with nonlocal point constraints for traffic flow through bottlenecks, such as exits in the context of pedestrians traffic and reduction of lanes on a road under construction in vehicular traffic. Constraints are defined based on data collected from non-local in space and/or in time observations of the flow. We propose a theoretical analysis and discretization framework that permits to include different data acquisition strategies; a numerical comparison is provided. Nonlocal constraint allows to model, e.g., the irrational behavior (" panic ") near the exit observed in dense crowds and the capacity drop at tollbooth in vehicular traffic. Existence and uniqueness of solutions are shown under suitable and " easy to check " assumptions on the constraint operator. A numerical scheme for the problem, based on finite volume methods, is designed, its convergence is proved and its validation is done with an explicit solution. Numerical examples show that nonlocally constrained models are able to reproduce important features in traffic flow such as self-organization.

yearjournalcountryeditionlanguage
2016-12-16Journal de Mathématiques Pures et Appliquées
https://doi.org/10.1016/j.matpur.2018.01.005