{"title":"A Hierarchy of Kinetic Discrete-Velocity Models for Traffic Flow Derived from a Nonlocal Prigogine–Herman Model","authors":"R. Borsche, A. Klar","doi":"10.1137/23m1583065","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 139-164, February 2024. <br/> Abstract. Starting from a nonlocal version of the Prigogine–Herman traffic model, we derive a natural hierarchy of kinetic discrete-velocity models for traffic flow consisting of systems of quasi-linear hyperbolic equations with relaxation terms. The hyperbolic main part of these models turns out to have several favorable features. In particular, we determine Riemann invariants and prove richness and total linear degeneracy of the hyperbolic systems. Moreover, a physically reasonable invariant domain is obtained for all equations of the hierarchy. Additionally, we investigate the full relaxation system with respect to stability and persistence of periodic (stop-and-go-type) solutions and derive a condition for the appearance of such solutions. Finally, numerical results for various situations are presented, illustrating the analytical findings.","PeriodicalId":51149,"journal":{"name":"SIAM Journal on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.9000,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1583065","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Applied Mathematics, Volume 84, Issue 1, Page 139-164, February 2024. Abstract. Starting from a nonlocal version of the Prigogine–Herman traffic model, we derive a natural hierarchy of kinetic discrete-velocity models for traffic flow consisting of systems of quasi-linear hyperbolic equations with relaxation terms. The hyperbolic main part of these models turns out to have several favorable features. In particular, we determine Riemann invariants and prove richness and total linear degeneracy of the hyperbolic systems. Moreover, a physically reasonable invariant domain is obtained for all equations of the hierarchy. Additionally, we investigate the full relaxation system with respect to stability and persistence of periodic (stop-and-go-type) solutions and derive a condition for the appearance of such solutions. Finally, numerical results for various situations are presented, illustrating the analytical findings.
期刊介绍：
SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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