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January  2020, 40(1): 233-266. doi: 10.3934/dcds.2020010

Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux

Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio 1, Coppito, I-67100 L'Aquila, Italy

Received  January 2019 Revised  May 2019 Published  October 2019

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density $ v(\rho) $ and a function of the space variable $ \phi(x) $. We cover four distinct cases in terms of the sign of $ \phi $, including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform $ BV $ estimate for the approximating density.

Citation: Marco Di Francesco, Graziano Stivaletta. Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 233-266. doi: 10.3934/dcds.2020010
References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.   Google Scholar
[2]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

A. Aw and M. Rascle, Resurrection of 'second order' models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916–938. doi: 10.1137/S0036139997332099.  Google Scholar

[4]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185–220. doi: 10.1007/s00205-007-0061-9.  Google Scholar

[5]

F. Berthelin and P. Goatin, Particle approximation of a constrained model for traffic flow, Nonlinear Differential Equations and Applications NoDEA, 24 (2017), Art. 55, 16 pp. doi: 10.1007/s00030-017-0480-8.  Google Scholar

[6]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855–885. doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[7]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds. CISM International Centre for Mechanical Sciences, 553 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1.  Google Scholar

[8]

J. A. Carrillo, M. Di Francesco and G. Toscani, Condensation phenomena in nonlinear drift equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 15 (2016), 145–171.  Google Scholar

[9]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[10]

M. Di Francesco, S. Fagioli and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, Journal of Differential Equations, 266 (2019), 2830–2868. doi: 10.1016/j.jde.2018.08.047.  Google Scholar

[11]

M. Di FrancescoS. Fagioli and M. D. Rosini, Many particle approximation for the Aw-Rascle-Zhang second order model for vehicular traffic, Mathematical Biosciences and Engineering, 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[12]

M. Di Francesco, S. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Bollettino dell'Unione Matematica Italiana, 10 (2017), 487–501. doi: 10.1007/s40574-017-0132-2.  Google Scholar

[13]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Deterministic particle approximation of the Hughes model in one space dimension, Kinetic and Related Models, 10 (2017), 215–237. doi: 10.3934/krm.2017009.  Google Scholar

[14]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, Active Particles, 1 (2017), 333–378.  Google Scholar

[15]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831–871. doi: 10.1007/s00205-015-0843-4.  Google Scholar

[16]

S. Fagioli and E. Radici, Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 1801–1829. doi: 10.1142/S0218202518400067.  Google Scholar

[17]

P. A. Ferrari, Shock fluctuations in asymmetric simple exclusion, Probabilty Theory and Related Fields, 91 (1992), 81–101. doi: 10.1007/BF01194491.  Google Scholar

[18]

P. L. Ferrari and P. Nejjar, Shock fluctuations in flat TASEP under critical scaling, Journal of Statistical Physics, 160 (2015), 985–1004. doi: 10.1007/s10955-015-1208-y.  Google Scholar

[19]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590–2606. doi: 10.1137/040608672.  Google Scholar

[20]

M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions, Communications in Mathematical Physics, 118 (1988), 31–59. doi: 10.1007/BF01218476.  Google Scholar

[21]

H. Holden and N. H. Risebro, The continuum limit of follow-the-leader models. a short proof, Discrete & Continuous Dynamical Systems-A, 38 (2018), 715–722. doi: 10.3934/dcds.2018031.  Google Scholar

[22]

H. Holden and N. H. Risebro, Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Networks & Heterogeneous Media, 13 (2018), 409–421. doi: 10.3934/nhm.2018018.  Google Scholar

[23]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques et Applications, Springer, 1997.  Google Scholar

[24]

K. H. Karlsen and K.-A. Lie, An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA Journal of Numerical Analysis, 19 (1999), 609–635. doi: 10.1093/imanum/19.4.609.  Google Scholar

[25]

K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete and Continuous Dynamical Systems, 9 (2003), 1081–1104. doi: 10.3934/dcds.2003.9.1081.  Google Scholar

[26]

S. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217–243. Google Scholar

[27]

T. M. Liggett, Interacting Particle Systems, vol. 276 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8542-4.  Google Scholar

[28]

M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 229 (1955), 317–345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[29]

D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[30]

P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42–51. doi: 10.1287/opre.4.1.42.  Google Scholar

[31]

M. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.  Google Scholar

[32]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2 (2003), 395–431.  Google Scholar

[33]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697–733. doi: 10.1002/cpa.3160430602.  Google Scholar

[34]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[35]

G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Communications in Partial Differential Equations, 37 (2012), 77–87. doi: 10.1080/03605302.2011.592236.  Google Scholar

[36]

C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

show all references

References:
[1] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.   Google Scholar
[2]

A. AwA. KlarT. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM Journal on Applied Mathematics, 63 (2002), 259-278.  doi: 10.1137/S0036139900380955.  Google Scholar

[3]

A. Aw and M. Rascle, Resurrection of 'second order' models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916–938. doi: 10.1137/S0036139997332099.  Google Scholar

[4]

F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and evolution of traffic jams, Archive for Rational Mechanics and Analysis, 187 (2008), 185–220. doi: 10.1007/s00205-007-0061-9.  Google Scholar

[5]

F. Berthelin and P. Goatin, Particle approximation of a constrained model for traffic flow, Nonlinear Differential Equations and Applications NoDEA, 24 (2017), Art. 55, 16 pp. doi: 10.1007/s00030-017-0480-8.  Google Scholar

[6]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855–885. doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[7]

J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds. CISM International Centre for Mechanical Sciences, 553 (2014), 1–46. doi: 10.1007/978-3-7091-1785-9_1.  Google Scholar

[8]

J. A. Carrillo, M. Di Francesco and G. Toscani, Condensation phenomena in nonlinear drift equations, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 15 (2016), 145–171.  Google Scholar

[9]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehrender Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.  Google Scholar

[10]

M. Di Francesco, S. Fagioli and E. Radici, Deterministic particle approximation for nonlocal transport equations with nonlinear mobility, Journal of Differential Equations, 266 (2019), 2830–2868. doi: 10.1016/j.jde.2018.08.047.  Google Scholar

[11]

M. Di FrancescoS. Fagioli and M. D. Rosini, Many particle approximation for the Aw-Rascle-Zhang second order model for vehicular traffic, Mathematical Biosciences and Engineering, 14 (2017), 127-141.  doi: 10.3934/mbe.2017009.  Google Scholar

[12]

M. Di Francesco, S. Fagioli and M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Bollettino dell'Unione Matematica Italiana, 10 (2017), 487–501. doi: 10.1007/s40574-017-0132-2.  Google Scholar

[13]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Deterministic particle approximation of the Hughes model in one space dimension, Kinetic and Related Models, 10 (2017), 215–237. doi: 10.3934/krm.2017009.  Google Scholar

[14]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows, Active Particles, 1 (2017), 333–378.  Google Scholar

[15]

M. Di Francesco and M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Archive for Rational Mechanics and Analysis, 217 (2015), 831–871. doi: 10.1007/s00205-015-0843-4.  Google Scholar

[16]

S. Fagioli and E. Radici, Solutions to aggregation–diffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Mathematical Models and Methods in Applied Sciences, 28 (2018), 1801–1829. doi: 10.1142/S0218202518400067.  Google Scholar

[17]

P. A. Ferrari, Shock fluctuations in asymmetric simple exclusion, Probabilty Theory and Related Fields, 91 (1992), 81–101. doi: 10.1007/BF01194491.  Google Scholar

[18]

P. L. Ferrari and P. Nejjar, Shock fluctuations in flat TASEP under critical scaling, Journal of Statistical Physics, 160 (2015), 985–1004. doi: 10.1007/s10955-015-1208-y.  Google Scholar

[19]

L. Gosse and G. Toscani, Identification of asymptotic decay to self-similarity for one-dimensional filtration equations, SIAM Journal on Numerical Analysis, 43 (2006), 2590–2606. doi: 10.1137/040608672.  Google Scholar

[20]

M. Z. Guo, G. C. Papanicolaou and S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions, Communications in Mathematical Physics, 118 (1988), 31–59. doi: 10.1007/BF01218476.  Google Scholar

[21]

H. Holden and N. H. Risebro, The continuum limit of follow-the-leader models. a short proof, Discrete & Continuous Dynamical Systems-A, 38 (2018), 715–722. doi: 10.3934/dcds.2018031.  Google Scholar

[22]

H. Holden and N. H. Risebro, Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow, Networks & Heterogeneous Media, 13 (2018), 409–421. doi: 10.3934/nhm.2018018.  Google Scholar

[23]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, vol. 26 of Mathématiques et Applications, Springer, 1997.  Google Scholar

[24]

K. H. Karlsen and K.-A. Lie, An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA Journal of Numerical Analysis, 19 (1999), 609–635. doi: 10.1093/imanum/19.4.609.  Google Scholar

[25]

K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete and Continuous Dynamical Systems, 9 (2003), 1081–1104. doi: 10.3934/dcds.2003.9.1081.  Google Scholar

[26]

S. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217–243. Google Scholar

[27]

T. M. Liggett, Interacting Particle Systems, vol. 276 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4613-8542-4.  Google Scholar

[28]

M. Lighthill and G. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 229 (1955), 317–345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[29]

D. Matthes and H. Osberger, Convergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation, ESAIM: Mathematical Modelling and Numerical Analysis, 48 (2014), 697-726.  doi: 10.1051/m2an/2013126.  Google Scholar

[30]

P. Richards, Shock waves on the highway, Operations Research, 4 (1956), 42–51. doi: 10.1287/opre.4.1.42.  Google Scholar

[31]

M. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013. doi: 10.1007/978-3-319-00155-5.  Google Scholar

[32]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 2 (2003), 395–431.  Google Scholar

[33]

G. Russo, Deterministic diffusion of particles, Communications on Pure and Applied Mathematics, 43 (1990), 697–733. doi: 10.1002/cpa.3160430602.  Google Scholar

[34]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, vol. 233 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[35]

G. Toscani, Finite time blow up in Kaniadakis-Quarati model of Bose-Einstein particles, Communications in Partial Differential Equations, 37 (2012), 77–87. doi: 10.1080/03605302.2011.592236.  Google Scholar

[36]

C. Villani, Optimal Transport. Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

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