[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ skip to main content
article

A convergent scheme for Hamilton---Jacobi equations on a junction: application to traffic

Published: 01 March 2015 Publication History

Abstract

In this paper, we consider first order Hamilton---Jacobi (HJ) equations posed on a "junction", that is to say the union of a finite number of half-lines with a unique common point. For this continuous HJ problem, we propose a finite difference scheme and prove two main results. As a first result, we show bounds on the discrete gradient and time derivative of the numerical solution. Our second result is the convergence (for a subsequence) of the numerical solution towards a viscosity solution of the continuous HJ problem, as the mesh size goes to zero. When the solution of the continuous HJ problem is unique, we recover the full convergence of the numerical solution. We apply this scheme to compute the densities of cars for a traffic model. We recover the well-known Godunov scheme outside the junction point and we give a numerical illustration.

References

[1]
Achdou, Y., Camilli, F., Cutri, A., Tchou, N.: Hamilton---Jacobi equations on networks. In: World Congress, vol. 18, pp. 2577---2582 (2011)
[2]
Achdou, Y., Camilli, F., Cutri, A., Tchou, N.: Hamilton---Jacobi equations constrained on networks. NoDEA Nonlinear Differ. 20, 413---445 (2013)
[3]
Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of $$L^1$$L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201, 27---86 (2011)
[4]
Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973---2004 (2000)
[5]
Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications, vol. 17. Springer, Paris (1994)
[6]
Bokanowski, O., Cheng, Y., Shu, C.-W.: A discontinuous Galerkin solver for front propagation. SIAM J. Sci. Comput. 33, 923---938 (2011)
[7]
Bokanowski, O., Zidani, H.: Anti-dissipative schemes for advection and application to Hamilton---Jacobi---Bellman equations. J. Sci. Comput. 30, 1---33 (2007)
[8]
Bressan, A.: Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem. Oxford Lecture Series in Mathematics and its Applications, vol. 20. Oxford University Press, Oxford (2000)
[9]
Bretti, G., Natalini, R., Piccoli, B.: Fast algorithms for the approximation of a fluid-dynamic model on networks. Discrete Contin. Dyn. Syst. Ser. B 6, 427---448 (2006)
[10]
Bretti, G., Natalini, R., Piccoli, B.: Numerical algorithms for simulations of a traffic model on road networks. J. Comput. Appl. Math. 210, 71---77 (2007)
[11]
Bretti, G., Natalini, R., Piccoli, B.: A fluid-dynamic traffic model on road networks. Arch. Comput. Methods Eng. 14, 139---172 (2007)
[12]
Camilli, F., Marchi, C., Schieborn, D.: The vanishing viscosity limit for Hamilton---Jacobi equations on Networks. J. Differ. Equ. 254, 4122---4143 (2013)
[13]
Camilli, F., Festa, A., Schieborn, D.: An approximation scheme for a Hamilton---Jacobi equation defined on a network. Appl. Numer. Math. 73, 33---47 (2013)
[14]
Capuzzo Dolcetta, I.: On a discrete approximation of the Hamilton---Jacobi equation of dynamic programming. Appl. Math. Optim. 4, 367---377 (1983)
[15]
Cheng, Y., Shu, C.-W.: Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension. SIAM J. Numer. Anal. 47, 4044---4072 (2010)
[16]
Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11, 215---234 (1928)
[17]
Crandall, M.G., Lions, P.L.: Two approximations of solutions of Hamilton---Jacobi equations. Math. Comput. 43, 1---19 (1984)
[18]
Crandall, M.G., Ishii, H., Lions, P.L.: User's guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27, 1---67 (1992)
[19]
Dafermos, C.M.: Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften, vol. 325. Springer, Berlin (2000)
[20]
Falcone, M.: A numerical approach to the infinite horizon problem of deterministic control theory. Appl. Math. Optim. 15, 1---13 (1987)
[21]
Falcone, M., Ferretti, R.: Discrete time high-order schemes for viscosity solutions of Hamilton---Jacobi---Bellman equations. Numer. Math. 67, 315---344 (1994)
[22]
Flötteröd, G., Rohde, J.: Operational macroscopic modeling of complex urban intersections. Transp. Res. B 45, 903---922 (2011)
[23]
Garavello, M., Natalini, R., Piccoli, B., Terracina, A.: Conservation laws with discontinuous flux. Netw. Heterog. Media 2, 159---179 (2006)
[24]
Garavello, M., Piccoli, B.: Traffic Flow on Networks. AIMS Series on Applied Mathematics, vol. 1. American Institute of Mathematical Sciences (AIMS), Springfield (2006)
[25]
Garavello, M., Piccoli, B.: Conservation laws on complex networks. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1925---1951 (2009)
[26]
Godunov, S.K.: A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Math. Sb. 47, 271---290 (1959)
[27]
Godlewski, E., Raviart, P.A.: Hyperbolic Systems of Conservation Laws. Mathematics and Applications, vol. 3/4. Ellipses, Paris (1991)
[28]
Göttlich, S., Herty, M., Ziegler, U.: Numerical discretization of Hamilton---Jacobi equations on networks. Netw. Heterog. Media 8, 685---705 (2013)
[29]
Hu, C., Shu, C.-W.: A discontinuous Galerkin finite element method for Hamilton---Jacobi equations. SIAM J. Sci. Comput. 21, 666---690 (1999)
[30]
Imbert, C., Monneau, R., Zidani, H.: A Hamilton---Jacobi approach to junction problems and application to traffic flows. ESAIM Control Optim. Calc. Var. 19, 129---166 (2013)
[31]
Khoshyaran, M.M., Lebacque, J.P.: Internal state models for intersections in macroscopic traffic flow models. In: Proceedings of Traffic and Granular Flow 09 (2009, accepted)
[32]
Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. In: CBMS-NSF Regional Conference Series in Applied Mathematics (No. 11) (1987)
[33]
Lebacque, J.P.: Les modèles macroscopiques du trafic. Annales des Ponts 67, 28---45 (1993)
[34]
Lebacque, J.P.: The Godunov scheme and what it means for first order traffic flow models. In: Lesort, J.B. (ed.) 13th ISTTT Symposium, pp. 647---678. Elsevier, New York (1996)
[35]
Lebacque, J.P., Khoshyaran, M.M.: Macroscopic flow models (First order macroscopic traffic flow models for networks in the context of dynamic assignment). In: Patriksson, M., Labbé, M. (eds.) Transportation Planning, the State of the Art, pp. 119---140. Klüwer Academic Press, Dordrecht (2002)
[36]
Lebacque, J.P., Koshyaran, M.M.: First-order macroscopic traffic flow models: intersection modeling, network modeling. In: Mahmassani, H.S. (ed.) Proceedings of the 16th International Symposium on the Transportation and Traffic Theory, College Park, Maryland, USA, pp. 365---386. Elsevier, Oxford (2005)
[37]
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
[38]
Lighthill, M.J., Whitham, G.B.: On kinetic waves. II. Theory of traffic flows on long crowded roads. Proc. R. Soc. Lond. Ser. A 229, 317---345 (1955)
[39]
Osher, S., Shu, C.-W.: High order essentially non-oscillatory schemes for Hamilton---Jacobi equations. SIAM J. Numer. Anal. 28, 907---922 (1991)
[40]
Perthame, B.: Kinetic Formulation of Conservation Laws. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2002)
[41]
Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42---51 (1956)
[42]
Serre, D.: Systems of Conservation Laws I: Hyperbolicity, Entropies, Shock Waves. Cambridge University Press, Cambridge (1999)
[43]
Tampere, C., Corthout, R., Cattrysse, D., Immers, L.: A generic class of first order node models for dynamic macroscopic simulations of traffic flows. Transp. Res. B 45, 289---309 (2011)
[44]
Xu, Z., Shu, C.-W.: Anti-diffusive high order WENO schemes for Hamilton---Jacobi equations. Methods Appl. Anal. 12, 169---190 (2005)
[45]
Zhang, Y.-T., Shu, C.-W.: High order WENO schemes for Hamilton---Jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005---1030 (2003)

Cited By

View all
  • (2019)Error estimates for a finite difference scheme associated with Hamilton---Jacobi equations on a junctionNumerische Mathematik10.1007/s00211-019-01043-9142:3(525-575)Online publication date: 20-Jul-2019
  1. A convergent scheme for Hamilton---Jacobi equations on a junction: application to traffic

      Recommendations

      Comments

      Please enable JavaScript to view thecomments powered by Disqus.

      Information & Contributors

      Information

      Published In

      cover image Numerische Mathematik
      Numerische Mathematik  Volume 129, Issue 3
      March 2015
      203 pages

      Publisher

      Springer-Verlag

      Berlin, Heidelberg

      Publication History

      Published: 01 March 2015

      Author Tags

      1. 35F21
      2. 65M06
      3. 65M12
      4. 90B20

      Qualifiers

      • Article

      Contributors

      Other Metrics

      Bibliometrics & Citations

      Bibliometrics

      Article Metrics

      • Downloads (Last 12 months)0
      • Downloads (Last 6 weeks)0
      Reflects downloads up to 19 Feb 2025

      Other Metrics

      Citations

      Cited By

      View all
      • (2019)Error estimates for a finite difference scheme associated with Hamilton---Jacobi equations on a junctionNumerische Mathematik10.1007/s00211-019-01043-9142:3(525-575)Online publication date: 20-Jul-2019

      View Options

      View options

      Figures

      Tables

      Media

      Share

      Share

      Share this Publication link

      Share on social media