Formulation of a maximum principle satisfying a numerical scheme for traffic flow models
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ORIGINAL PAPER
Formulation of a maximum principle satisfying a numerical scheme for traffic flow models Oluwaseun Farotimi1 • Kuppalapalle Vajravelu1 Received: 12 March 2020 / Accepted: 16 July 2020 Springer Nature Switzerland AG 2020
Abstract We consider a non-local traffic flow model with Arrhenius look-ahead dynamics. In recent times, a maximum principle satisfying local conservation framework has been getting much attention, yet conventional numerical approximation scheme may lead to a breakdown of the maximum principle. In this paper, we construct a maximum principle satisfying a numerical scheme for a class of non-local conservation laws and present numerical simulations for the traffic flow models. The technique and the idea developed in this work are applicable to a large class of non-local conservation laws. Keywords Nonlocal conservation law Traffic flow Maximum principle Numerical scheme Mathematics Subject Classification Primary 65M06; Secondary 35L65
1 Introduction In this work, we investigate numerically a class of non-local conservation laws [4], ot u þ ox Fðu; uÞ ¼ 0; t [ 0; x 2 R; ð1:1Þ uð0; xÞ ¼ u0 ðxÞ; x 2 R; where u is the unknown, F is a given smooth function and u is given by Z uðt; xÞ ¼ ðK uÞðt; xÞ ¼ Kðx yÞuðt; yÞdy
ð1:2Þ
R
where K is to be chosen later but K 2 W 1;1 ðRÞ. The history of mathematical theory for traffic flow goes far back to 1920, when Frank Knight pioneered an analysis for traffic equilibrium [19]. Traffic behavior depends on the interactions of vehicles in a complex-nonlinear way, individual reactions of human drivers This article is part of the section ‘‘Theory ofPDEs’’ edited by Eduardo Teixeira. & Kuppalapalle Vajravelu [email protected] 1
Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364, USA SN Partial Differential Equations and Applications
20 Page 2 of 11
SN Partial Differ. Equ. Appl. (2020)1:20
etc. Vehicles interaction is not just by obeying the laws of mechanics but rather cluster formation and shock wave propagation (forward and backward), based on the vehicle density [16, 18–19]. Some of the current traffic models uses a mixture of empirical relations and theoretical techniques where the models are then developed into traffic predictions, taking into account changes such as increased vehicle use, changes in land use or modes of transport etc. [6, 9, 12]. Current researchers have extended the findings of the traffic flow theory. Robust and reliable numerical methods for deterministic traffic flow have been studied in [1]. Local theory and blowup analysis of conservation law with nonlocal flux have been studied in [3]. A class of nonlocal conservation laws where the nonlinear advection coupled with both local and nonlocal mechanisms have been studied in [2]. Also, Riccati-type system that governs the flow gradient was investigated in [13]. The applications of the class of conservation laws in [1] have its place in radiating gas motion, dispersive water wav
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