While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,...

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 stop-and-go 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 self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well. CONTENTS

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code

The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasi-continuum method.

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We develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order one 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 can not 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 second order scheme that works effectively, with a fixed spatial and temporal discretization, for all range of mean free path. Formal uniform consistency proof for a first order scheme, and numerical convergence proof for the second order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivate...

Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long-time behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid. We introduce a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers). The idea is to build into the numerical scheme the asymptotic balances that lead to this behavior. Numerical experiments on 2 \Theta 2 systems verify our analysis. 1 E-mail address: jin@math.gatech.edu 2 E-mail address: lvrmr@math.arizona.edu Typeset by A M S-T E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...

. Convergence is established for a scalar nite dierence scheme, based on the Godunov or Engquist-Osher 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 Kruzkov-type 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...

A new method for the computation of conserved densities of nonlinear dierentialdi erence equations is applied to Toda lattices and discretizations of the Kortewegde Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability. Keywords: Conserved densities; Integrability; Semi-discrete equations; Lattice 1 Introduction Nonlinear dierential-dierence equations (DDEs) describe many interesting phenomena such as vibrations of particles in lattices, charge uctuations in networks, Langmuir waves in plasmas, interactions between competing populations. Mathematically, DDEs also occur as spatially discrete analogues of partial dierential equations (PDEs). As such, lattices play a key role in numerical solvers for PDEs [1]. In [2-4], we introduced an algorithm to nd the analytical form of polynomial conserved densities for systems of nonlinear evolution equations. We use...

Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small-relaxation limit governed by reduced systems of parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which thus are able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.