Averaged kinetic models for flows on unstructured networks

Kinetic and Related Models

Volumn 4, Issue 4, 2011, Pages 1081-1096

  Herty, Michael ^a   Ringhofer, Christian ^b  

^a RWTH AACHEN UNIVERSITY   (Germany)

^b ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Asymptotic analysis; Kinetic theory; Network dynamics

Indexed keywords

EID: 84856252938     PISSN: 19375093     EISSN: 19375077     Source Type: Journal    
DOI: 10.3934/krm.2011.4.1081     Document Type: Article

References (21)

1. D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920.
2. A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.
3. S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks, Journal of Economic Dynamics and Control, 31 (2007), 2061-2084.
4. D. Brockmann, Anomalous diffusion and the structure of human transportation networks, Eur. Phys. J. Special Topics, 157 (2008), 173-189.
5. C. Cercignani, I. Gamba and D. Levermore, A drift-collision balance for a Boltzmann-Poisson system in bounded domains, SIAM J. Appl. Math., 61 (2001), 1932-1958.
6. C. Cercignani, R. Illner and M. Pulvirenti, " The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.
7. F. Della Rossa, C. D'Angelo and A. Quarteroni, A distributed model of traffic flows on extended regions, Networks and Heterogeneous Media 5 (2010), 525-544.
8. P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random break-downs, SIAM J. Applied Mathematics, 68 (2007), 59-79.
9. A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256.
10. M. Garavello and B. Piccoli, " Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, Spring-field, MO, 2006.
11. S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks, European Journal of Operational Research, 202 (2010), 456-465.
12. S. Göttlich, M. Herty and C. Ringhofer, Time-dependent order and distribution policies in supply networks, in " Progress in Industrial Mathematics at ECMI 2008," Mathematics in Industry, Vol. 15 (eds, J. Norbury, H. Ockendon and E. Wilson), 521-526, Springer, 2010.
13. W. H. S. C. Graves and D. B. Kletter, A dynamic model for requirements planning with application to supply chain optimization, Operations Research, 46 (1998), 35-49.
14. D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141.
15. E. W. Larsen, A generalized Boltzmann equation for "non-classical" particle transport, in " Joint International Topical Meeting on Mathematics and Computation and Supercomputing in Nuclear Applications" (M& C + SNA 2007), Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2007.
16. E. W. Larsen and R. Vasques, A generalized linear Boltzmann equation for non-classical particle transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 2010.
17. M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
18. H. Missbauer, Aggregate order release planning for time varying demand, Int. J. Production Research, 40 (2002), 699-718.
19. K. Nagel, Particle hopping models and traffic flow theory, Physical Review E, 3 (1996), 4655-4672.
20. F. Poupaud, Runaway phenomena and fluid approximation under high fields in semicon- ductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372.
21. D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442.