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Traffic and related selfdriven manyparticle systems
, 2000
"... Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? ..."
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Cited by 187 (30 self)
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Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? What are the mechanisms behind stopandgo traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ‘‘freeze by heating’’? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to selfdriven manyparticle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particlebased), mesoscopic (gaskinetic), and macroscopic (fluiddynamic) models. Attention is also paid to the formulation of a micromacro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for selfdriven manyparticle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socioeconomic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.
The EulerPoincaré equations and semidirect products with applications to continuum theories
 ADV. MATH
, 1998
"... We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. ..."
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Cited by 149 (66 self)
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We study Euler–Poincaré systems (i.e., the Lagrangian analogue of LiePoisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d’Alembert type. Then we derive an abstract KelvinNoether theorem for these equations. We also explore their relation with the theory of LiePoisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional CamassaHolm equations, which have many potentially interesting analytical properties. These
Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy
 Comm. Pure Appl. Math
, 1992
"... We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N \Theta N systems that ..."
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Cited by 143 (8 self)
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We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N \Theta N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 \Theta 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 \Theta 2 case. 1 Email address: cheng@zaphod.uchicago.edu 2 Email address: lvrmr@math.arizona.edu 3 Email address: liu@pde.stanford.edu 2 1. Introduction We are concerned with the phenomena of relaxation, particularly the question of stability and singular limits of zero relaxation time. Relaxation is import...
Statistical physics of vehicular traffic and some related systems
 PHYSICS REPORT 329
, 2000
"... ..."
formalism of onedimensional systems of hydrodynamic type and the Bogolyubov  Whitham averaging method
 Soviet Math. Dokl
, 1983
"... I. Onedimensional systems of hydrodynamic type. Poisson brackets and Riemannian geometry. From a purely mathematical point of view many systems such as ideal fluids including mixtures and systems with internal degrees of freedom are given in the onedimensional case by equations of first order (1) ..."
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Cited by 80 (6 self)
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I. Onedimensional systems of hydrodynamic type. Poisson brackets and Riemannian geometry. From a purely mathematical point of view many systems such as ideal fluids including mixtures and systems with internal degrees of freedom are given in the onedimensional case by equations of first order (1) uit = v i j(u)u j x. The field variables ui are usually the density of momentum and energy (or mass) and possibly a number of others. The Euler equations for ideal fluids are Hamiltonian with respect to Poisson brackets of special form (see [1] and [2]) which in the onedimensional case we shall generalize to general systems of the form (1) in connection with new applications. Definition 1. A Poisson bracket on a space of fields uk(x) is called a bracket of hydrodynamic type if it has the form (2) {ui(x), uj(y)} = gij(u(x))δ′(x − y) + bijk (u(x))ukxδ(x − y). For any pair of functionals I =
Resurrection of "Second Order" Models of Traffic Flow?
 SIAM J. APPL. MATH
, 1999
"... We introduce a new "second order" model of traffic flow. As noted in [2], the previous "second order" models i.e models with two equations (mass and "momentum") lead to nonphysical effects, probably because they try to mimic the gas dynamics equations, with an unrealist ..."
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Cited by 78 (6 self)
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We introduce a new "second order" model of traffic flow. As noted in [2], the previous "second order" models i.e models with two equations (mass and "momentum") lead to nonphysical effects, probably because they try to mimic the gas dynamics equations, with an unrealistic dependence of the acceleration with respect to the space derivative of the "pressure". We simply replace this space derivative by a convective derivative, and we show that this very simple repair completely resolves the inconsistencies of these models. Moreover, our model nicely predicts instabilities nearby the vacuum i.e. for a very light traffic.
Convergence to Equilibrium for the Relaxation Approximations of Conservation Laws
, 1996
"... We study the Cauchy problem for 2\Theta2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of ..."
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Cited by 78 (13 self)
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We study the Cauchy problem for 2\Theta2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to zero. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. 1. Introduction In this paper we are interested to the relaxation behaviour of the following system of hyperbolic conservation laws with a singular perturbation source (1.1) ae @ t u + @ x v = 0 ; @ t v + @ x oe(u) = \Gamma 1 " (v \Gamma f(u)) (" ? 0); for (x; t) 2 IR \Theta (0; 1). Here oe, f are some given smooth functions such that oe 0 (u) ( ? 0), f(0) = 0. The system (1.1) is equivalent to the onedimensional perturbed wave equation (1.2) @ tt w \Gamma @ x oe(@ x...
The Curvelet Representation of Wave Propagators is Optimally Sparse
, 2004
"... This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shape ..."
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Cited by 68 (13 self)
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This paper argues that curvelets provide a powerful tool for representing very general linear symmetric systems of hyperbolic differential equations. Curvelets are a recently developed multiscale system [10, 7] in which the elements are highly anisotropic at fine scales, with effective support shaped according to the parabolic scaling principle width ≈ length 2 at fine scales. We prove that for a wide class of linear hyperbolic differential equations, the curvelet representation of the solution operator is both optimally sparse and well organized. • It is sparse in the sense that the matrix entries decay nearly exponentially fast (i.e. faster than any negative polynomial), • and wellorganized in the sense that the very few nonnegligible entries occur near a few shifted diagonals. Indeed, we show that the wave group maps each curvelet onto a sum of curveletlike waveforms whose locations and orientations are obtained by following the different Hamiltonian flows—hence the diagonal shifts in the curvelet representation. A physical interpretation of this result is that curvelets may be viewed as coherent waveforms with enough frequency localization so that they behave like waves but at the same time, with enough spatial localization so that they simultaneously behave like particles.
Uniformly accurate schemes for hyperbolic systems with relaxations
 SIAM J. Numer. Anal
, 1997
"... Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underreso ..."
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Cited by 65 (21 self)
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Abstract. We develop highresolution shockcapturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one cannot decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a secondorder scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a firstorder scheme and numerical convergence proof for the secondorder scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.
Convergence Of A Difference Scheme For Conservation Laws With A Discontinuous Flux
 SIAM J. Numer. Anal
, 1999
"... . Convergence is established for a scalar nite dierence scheme, based on the Godunov or EngquistOsher ux, for scalar conservation laws having a ux that is spatially dependent through a possibly discontinuous coecient. Other works in this direction have established convergence for methods employing ..."
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Cited by 64 (8 self)
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. Convergence is established for a scalar nite dierence scheme, based on the Godunov or EngquistOsher ux, for scalar conservation laws having a ux that is spatially dependent through a possibly discontinuous coecient. Other works in this direction have established convergence for methods employing the solution of 2x2 Riemann problems. The algorithm discussed here uses only scalar Riemann solvers. Satisfaction of a set of Kruzkovtype entropy inequalities is established for the limit solution, from which geometric entropy conditions follow. Assuming a piecewise constant coecient, it is shown that these conditions imply L 1 contractiveness for piecewise C 1 solutions, thus extending a well known theorem. Key words. conservation laws, dierence approximations, discontinuous coecients AMS subject classications. 35L65, 65M06, 65M12, 35R05 1. Introduction. The subject of this paper is a nite dierence algorithm for computing approximate solutions of the Cauchy problem for scala...