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68
A NonOscillatory Eulerian Approach to Interfaces in Multimaterial Flows (The Ghost Fluid Method)
, 2000
"... While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,... ..."
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Cited by 167 (41 self)
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While Eulerian schemes work well for most gas flows, they have been shown to admit nonphysical oscillations near some material interfaces. In contrast,...
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 146 (23 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.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 125 (17 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi 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 selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
ANALYSIS OF MULTISCALE METHODS
, 2004
"... 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 quasicontinuum method. ..."
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Cited by 118 (13 self)
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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 quasicontinuum method.
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
"... ..."
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 61 (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.
Numerical Schemes For Hyperbolic Conservation Laws With Stiff Relaxation Terms
 J. Comput. Phys
, 1996
"... Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime 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 meth ..."
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Cited by 57 (11 self)
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Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime 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 Email address: jin@math.gatech.edu 2 Email address: lvrmr@math.arizona.edu Typeset by A M ST E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...
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 47 (4 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...
Computation of Conserved Densities for Systems of Nonlinear DifferentialDifference Equations
, 1997
"... 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, ..."
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Cited by 40 (22 self)
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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; Semidiscrete equations; Lattice 1 Introduction Nonlinear dierentialdierence 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 [24], we introduced an algorithm to nd the analytical form of polynomial conserved densities for systems of nonlinear evolution equations. We use...
Fully adaptive multiresolution finite volume schemes for conservation laws
 Math. Comp
, 2003
"... Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at ..."
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Cited by 26 (13 self)
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Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity. 1.