Averaged kinetic models for flows onunstructured networks

Michael Herty; Christian Ringhofer

doi:10.3934/krm.2011.4.1081

Article Contents

2011, Volume 4, Issue 4: 1081-1096. Doi: 10.3934/krm.2011.4.1081

This issue Previous Article The Spherical Harmonics Expansion model coupled to the Poisson equation Next Article A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping

Averaged kinetic models for flows on unstructured networks

Michael Herty^1, and
Christian Ringhofer^2,

1.
RWTH Aachen, Department of Mathematics, Templergraben 55, 52056 Aachen
2.
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804

Received: June 30, 2011

Revised: August 31, 2011

Published: November 2011

Abstract Related Papers Cited by

Abstract

We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[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.doi: 10.1137/040604625.
[2]	A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512.doi: 10.1126/science.286.5439.509.
[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.doi: 10.1016/j.jedc.2007.01.004.
[4]	D. Brockmann, Anomalous diffusion and the structure of human transportation networks, Eur. Phys. J. Special Topics, 157 (2008), 173-189.doi: 10.1140/epjst/e2008-00640-0.
[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.doi: 10.1137/S0036139999360465.
[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 breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79.doi: 10.1137/060674302.
[9]	A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted aver- ages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256.doi: 10.1093/imanum/drm050.
[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, Springfield, 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.doi: 10.1016/j.ejor.2009.05.028.
[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.doi: 10.1287/opre.46.3.S35.
[14]	D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141.doi: 10.1103/RevModPhys.73.1067.
[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.doi: 10.1016/j.jqsrt.2010.07.003.
[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.doi: 10.1080/00207540110090939.
[19]	K. Nagel, Particle hopping models and traffic flow theory, Physical Review E, 3 (1996), 4655-–4672.doi: 10.1103/PhysRevE.53.4655.
[20]	F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372.doi: 10.1002/zamm.19920720813.
[21]	D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442.doi: 10.1038/30918.