Abstract
In the context of road traffic modeling we consider a scalar hyperbolic conservation law with the flux (fundamental diagram) which is discontinuous at \(x=0\), featuring variable velocity limitation. The flow maximization criterion for selection of a unique admissible weak solution is generally admitted in the literature, however justification for its use can be traced back to the irrelevant vanishing viscosity approximation. We seek to assess the use of this criterion on the basis of modeling proper to the traffic context. We start from a first order microscopic follow-the-leader (FTL) model deduced from basic interaction rules between cars. We run numerical simulations of FTL model with large number of agents on truncated Riemann data, and observe convergence to the flow-maximizing Riemann solver. As an obstacle towards rigorous convergence analysis, we point out the lack of order-preservation of the FTL semigroup.
MDR is member of GNAMPA. MDR acknowledges the support of the National Science Centre, Poland, Project “Mathematics of multi-scale approaches in life and social sciences” No. 2017/25/B/ST1/00051, by the INdAM-GNAMPA Project 2019 “Equazioni alle derivate parziali di tipo iperbolico o non locale ed applicazioni” and by University of Ferrara, FIR Project 2019 “Leggi di conservazione di tipo iperbolico: teoria ed applicazioni”.
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References
Adimurthi, Mishra, S., Gowda, G.D.V.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2(4), 783–837 (2005)
Amadori, D.: Initial-boundary value problems for nonlinear systems of conservation laws. NoDEA Nonlinear Differ. Equ. Appl. 4(1), 1–42 (1997)
Andreianov, B.: The semigroup approach to conservation laws with discontinuous flux. In: Hyperbolic Conservation Laws and Related Analysis with Applications, Springer Proceedings in Mathematics and Statistics, vol. 49, pp. 1–22. Springer, Heidelberg (2014)
Andreianov, B.: New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux. In: CANUM 2014–42e Congrès National d’Analyse Numérique, ESAIM Proceedings Surveys, vol. 50, pp. 40–65. EDP Sci, Les Ulis (2015)
Andreianov, B., Cancès, C.: Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium. Comput. Geosci. 17(3), 551–572 (2013)
Andreianov, B., Cancès, C.: On interface transmission conditions for conservation laws with discontinuous flux of general shape. J. Hyperbolic Differ. Equ. 12(2), 343–384 (2015)
Andreianov, B., Donadello, C., Razafison, U., Rosini, M.D.: Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks. ESAIM Math. Model. Numer. Anal. 50(5), 1269–1287 (2016)
Andreianov, B., Donadello, C., Razafison, U., Rosini, M.D.: Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux. J. Math. Pures Appl. 9(116), 309–346 (2018)
Andreianov, B., Donadello, C., Rosini, M.D.: Crowd dynamics and conservation laws with nonlocal constraints and capacity drop. Math. Models Methods Appl. Sci. 24(13), 2685–2722 (2014)
Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of \(L^1\)-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201(1), 27–86 (2011)
Andreianov, B., Mitrović, D.: Entropy conditions for scalar conservation laws with discontinuous flux revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(6), 1307–1335 (2015)
Andreianov, B., Sbihi, K.: Well-posedness of general boundary-value problems for scalar conservation laws. Trans. Am. Math. Soc. 367(6), 3763–3806 (2015)
Bardos, C., le Roux, A.Y., Nédélec, J.C.: First order quasilinear equations with boundary conditions. Commun. Partial Differ. Equ. 4(9), 1017–1034 (1979)
Benyahia, M., Rosini, M.D.: A macroscopic traffic model with phase transitions and local point constraints on the flow. Netw. Heterog. Media 12(2), 297–317 (2017)
Braess, D.: Über ein paradoxon aus der verkehrsplanung. Unternehmensforschung 12(1), 258–268 (1968)
Bürger, R., Karlsen, K., Risebro, N., Towers, J.: Monotone difference approximations for the simulation of clarifier-thickener units. Comput. Vis. Sci. 6(2), 83–91 (2004)
Bürger, R., Karlsen, K.H., Towers, J.D.: An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47(3), 1684–1712 (2009)
Bürger, R., Karlsen, K.H., Towers, J.D.: On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Netw. Heterog. Media 5(3), 461–485 (2010)
Cancès, C.: Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution. SIAM J. Math. Anal. 42(2), 946–971 (2010)
Colombo, R.M., Garavello, M.: Phase transition model for traffic at a junction. J. Math. Sci. (N.Y.) 196(1), 30–36 (2014)
Colombo, R.M., Goatin, P.: A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234(2), 654–675 (2007)
Colombo, R.M., Rosini, M.D.: Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28(13), 1553–1567 (2005)
Colombo, R.M., Rosini, M.D.: Well posedness of balance laws with boundary. J. Math. Anal. Appl. 311(2), 683–702 (2005)
Corli, A., Rosini, M.D.: Coherence and chattering of a one-way valve. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik (2019)
Dal Santo, E., Donadello, C., Pellegrino, S.F., Rosini, M.D.: Representation of capacity drop at a road merge via point constraints in a first order traffic model. ESAIM Math. Model. Numer. Anal. 53(1), 1–34 (2019)
Delle Monache, M.L., Goatin, P., Piccoli, B.: Priority-based Riemann solver for traffic flow on networks. Commun. Math. Sci. 16(1), 185–211 (2018)
Di Francesco, M., Fagioli, S., Rosini, M.D.: Deterministic particle approximation of scalar conservation laws. Boll. Unione Mat. Ital. 10(3), 487–501 (2017)
Di Francesco, M., Fagioli, S., Rosini, M.D., Russo, G.: Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows. In: Active Particles. vol. 1. Advances in Theory, Models, and Applications, pp. 333–378. Model. Simul. Sci. Eng. Technol. Birkhäuser/Springer, Cham (2017)
Di Francesco, M., Rosini, M.: Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Arch. Rat. Mech. Anal. 217(3), 831–871 (2015)
Di Francesco, M., Fagioli, S., Rosini, M.D.: Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Math. Biosci. Eng. 14(1), 127–141 (2017)
Di Francesco, M., Fagioli, S., Rosini, M.D., Russo, G.: Deterministic particle approximation of the Hughes model in one space dimension. Kinet. Relat. Models 10(1), 215–237 (2017)
Diehl, S.: Continuous sedimentation of multi-component particles. Math. Methods Appl. Sci. 20(15), 1345–1364 (1997)
Diehl, S.: A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 6(1), 127–159 (2009)
Garavello, M., Goatin, P.: The Aw-Rascle traffic model with locally constrained flow. J. Math. Anal. Appl. 378(2), 634–648 (2011)
Garavello, M., Natalini, R., Piccoli, B., Terracina, A.: Conservation laws with discontinuous flux. Netw. Heterog. Media 2(1), 159–179 (2007)
Gimse, T., Risebro, N.H.: Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23(3), 635–648 (1992)
Hopf, E.: The partial differential equation \(u_t+uu_x=\mu u_{xx}\). Commun. Pure Appl. Math. 3, 201–230 (1950)
Kaasschieter, E.F.: Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium. Comput. Geosci. 3(1), 23–48 (1999)
Karlsen, K.H., Risebro, N.H., Towers, J.D.: \(L^1\) stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3, 1–49 (2003)
Kolb, O., Costeseque, G., Goatin, P., Göttlich, S.: Pareto-optimal coupling conditions for the Aw-Rascle-Zhang traffic flow model at junctions. SIAM J. Appl. Math. 78(4), 1981–2002 (2018)
Kruzhkov, S.N.: First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (123), 228–255 (1970)
Lax, P.: Shock Waves and Entropy, pp. 603–634 (1971)
LeFloch, P.G.: Hyperbolic Systems of Conservation Laws. Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2002)
Lighthill, M., Whitham, G.: On kinematic waves. II. A theory of traffic flow on long crowded roads. In: Royal Society of London. Series A, Mathematical and Physical Sciences. vol. 229, pp. 317–345 (1955)
Moutari, S., Herty, M., Klein, A., Oeser, M., Steinauer, B., Schleper, V.: Modelling road traffic accidents using macroscopic second-order models of traffic flow. IMA J. Appl. Math. 78(5), 1087–1108 (2013)
Oleĭnik, O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspehi Mat. Nauk 14(2(86)), 165–170 (1959)
Rayleigh, L.: Aerial plane waves of finite amplitude [Proc. Roy. Soc. London Ser. A 84 (1910), 247–284]. In: Classic papers in shock compression science, pp. 361–404. High-press. Shock Compression Condens. Matter, Springer, New York (1998)
Richards, P.I.: Shock waves on the highway. Oper. Res. 4(1), 42–51 (1956)
Ross, D.S.: Two new moving boundary problems for scalar conservation laws. Commun. Pure Appl. Math. 41(5), 725–737 (1988)
Seguin, N., Vovelle, J.: Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13(2), 221–257 (2003)
Shen, W.: Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition. Netw. Heterog. Media 13(3), 449–478 (2018)
Srivastava, A., Geroliminis, N.: Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model. Transp. Res. Part C: Emerg. Technol. 30, 161–177 (2013)
Towers, J.D.: Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38(2), 681–698 (2000)
Villa, S., Goatin, P., Chalons, C.: Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model. Discrete Contin. Dyn. Syst. Ser. B 22(10), 3921–3952 (2017)
Vol\(^{\prime }\)pert, A.I.: Spaces \({\text{BV}}\) and quasilinear equations. Mat. Sb. (N.S.) 73 (115), 255–302 (1967)
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Andreianov, B., Rosini, M.D. (2020). Microscopic Selection of Solutions to Scalar Conservation Laws with Discontinuous Flux in the Context of Vehicular Traffic. In: Banasiak, J., Bobrowski, A., Lachowicz, M., Tomilov, Y. (eds) Semigroups of Operators – Theory and Applications. SOTA 2018. Springer Proceedings in Mathematics & Statistics, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-030-46079-2_7
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