Systems of conservation laws with discontinuous fluxes and applications to traffic

Massimiliano Rosini

Abstract


In this paper we study \(2\times 2\) systems of conservation laws with discontinuous fluxes arising in vehicular traffic modeling. The main goal is to introduce an appropriate notion of solution. To this aim we consider physically reasonable microscopic follow-the-leader models. Macroscopic Riemann solvers are then obtained as many particle limits. This approach leads us to develop six models. We propose a unified way to describe such models, which highlights their common property of maximizing the density flow across the interface under appropriate physical restrictions depending on the case at hand.

Keywords


Conservation laws; Aw-Rascle-Zhang model for vehicular traffic; discontinuous flux; follow-the-leader model; Riemann solvers; point constraint on the flux; point constraint on the velocity

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References


Adimurthi, Dutta, R., Ghoshal, S. S., Veerappa Gowda, G. D., Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math. 64 (1) (2011), 84–115.

Adimurthi, Dutta, R., Gowda, G. D. V., Jaffre, J., Monotone (A,B) entropy stable numerical scheme for scalar conservation laws with discontinuous flux, ESAIM Math. Model. Numer. Anal. 48 (6) (2014), 1725–1755.

Adimurthi, Mishra, S., Gowda, G. D. V., Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ. 2 (4) (2005), 783–837.

Adimurthi, Mishra, S., Gowda, G. D. V., Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media 2 (1) (2007), 127–157.

Andreianov, B., The semigroup approach to conservation laws with discontinuous flux, in: Hyperbolic conservation laws and related analysis with applications, Springer Proc. Math. Stat. 49, Springer, Heidelberg, 2014, 1–22.

Andreianov, B., New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, in: CANUM 2014 – 42e Congres National d’Analyse Numerique, ESAIM Proc. Surveys 50 EDP Sci., Les Ulis, 2015, 40–65.

Andreianov, B., Cances, C., Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium, Comput. Geosci. 17 (3) (2013), 551–572.

Andreianov, B., Cances, C., On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ. 12 (2) (2015), 343–384.

Andreianov, B., Donadello, C., Rosini, M. D., A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci. 26 (4) (2016), 751–802.

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) (2011), 27–86.

Andreianov, B., Mitrovic, D., Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincare Anal. Non Lineaire 32 (6) (2015), 1307–1335.

Andreianov, B., Rosini, M. D., Microscopic selection of solutions to scalar conservation laws with discontinuous flux in the context of vehicular traffic, submitted, 2019.

Aw, A., Rascle, M., Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math. 60 (3) (2000), 916–938.

Burger, R., Karlsen, K., Risebro, N., Towers., J., Monotone difference approximations for the simulation of clarifier-thickener units, Computing and Visualization in Science 6 (2) (2004), 83–91.

Burger, 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) (2009), 1684–1712.

Burger, 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) (2010), 461–485.

Cances, 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) (2010), 946–971.

Colombo, R. M., Goatin, P., A well posed conservation law with a variable unilateral constraint, J. Differential Equations 234 (2) (2007), 654–675.

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) (2017), 127–141.

Diehl, S., Continuous sedimentation of multi-component particles, Math. Methods Appl. Sci. 20 (15) (1997), 1345–1364.

Diehl, S., A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equ. 6 (1) (2009), 127–159.

Garavello, M., Natalini, R., Piccoli, B., Terracina, A., Conservation laws with discontinuous flux, Netw. Heterog. Media 2 (1) (2007), 159–179.

Ghoshal, S. S., Optimal results on TV bounds for scalar conservation laws with discontinuous flux, J. Differential Equations 258 (3) (2015), 980–1014.

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) (1992), 635–648.

Hopf, E., The partial differential equation (u_t+uu_x = mu u_{xx}), Comm. Pure Appl. Math. 3 (1950), 201–230.

Kaasschieter, E. F., Solving the Buckley–Leverett equation with gravity in a heterogeneous porous medium, Comput. Geosci. 3 (1) (1999), 23–48.

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 (2003), 49 pp.

Kruzhkov, S. N., First order quasilinear equations with several independent variables, Mat. Sb. (N. S.) 81 (123) (1970), 228–255.

Lighthill, M., Whitham, G., On kinematic waves II. A theory of traffic flow on long crowded roads, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 229 (1955), 317–345.

Rayleigh, L., Aerial plane waves of finite amplitude [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 84 (1910), 247–284], in: Classic Papers in Shock Compression Science, High-press, Shock Compression Condens. Matter, Springer, New York, 1998, 361–404.

Richards, P. I., Shock waves on the highway, Operations Research 4 (1) (1956), 42–51.

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) (2003), 221–257.

Shen, W., Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition, Netw. Heterog. Media 13 (3) (2018), 449–478.

Shen, W., Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads, Netw. Heterog. Media 14 (4) (2019), 709–732.

Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal. 38 (2) (2000), 681–698.

Zhang, H., A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological 36 (3) (2002), 275–290.




DOI: http://dx.doi.org/10.17951/a.2019.73.2.135-173
Date of publication: 2020-01-16 07:29:36
Date of submission: 2020-01-04 22:09:20


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