A model reduction technique based on an optimization principle is employed to coarse-grain inviscid, incompressible fluid dynamics in two dimensions. In this reduction the spectrally-truncated vorticity equation defines the microdynamics, while the macroscopic state space consists of quasi-equilibrium trial probability densities on the microscopic phase space, which are parameterized by the means and variances of the low modes of the vorticity. A macroscopic path therefore represents a coarse-grained approximation to the evolution of a nonequilibrium ensemble of microscopic solutions. Closure in terms of the vector of resolved variables, namely, the means and variances of the low modes, is achieved by minimizing over all feasible paths the time integral of their mean-squared residual with respect to the Liouville equation. The equations governing the optimal path are deduced from Hamilton-Jacobi theory. The coarse-grained dynamics derived by this optimization technique contains a scale-dependent eddy viscosity, modified nonlinear interactions between the low mode means, and a nonlinear coupling between the mean and variance of each low mode. The predictive skill of this optimal closure is validated quantitatively by comparing it against direct numerical simulations. These tests show that good agreement is achieved without adjusting any closure parameters.

We consider a two-dimensional (2D) counterpart of the experiment that led to the creation of quasi-1D bright solitons in Bose-Einstein condensates (BECs) (2002 Nature 417 150-3). We start by identifying the fundamental state of the 2D Gross-Pitaevskii equation for repulsive interactions, with a harmonic-oscillator (HO) trap, and with or without an optical lattice (OL). Subsequently, we switch the sign of the interaction to induce interatomic attraction and monitor the ensuing dynamics. Regions of stable self-trapping and a catastrophic collapse of 2D fundamental states are identified in the parameter plane of the OL strength and BEC norm. The increase of the OL strength expands the persistence domain for the solitary waves to larger norms. For single-charged solitary vortices, in addition to the survival and collapse regimes, an intermediate one is identified, where the vortex resists the collapse but loses its structure, transforming into a single-hump state. The same setting may also be implemented in the context of optical solitons and vortices, using photonic-crystal fibers.

We report results of systematic simulations of the dynamics of solitons in the framework of the one-dimensional nonlinear Schrödinger equation (NLSE), which includes the harmonic-oscillator (HO) potential and a random potential. The equation models experimentally relevant spatially disordered settings in Bose-Einstein condensates (BECs) and nonlinear optics. First, the generation of soliton arrays from a broad initial quasi-uniform state by the modulational instability (MI) is considered, following a sudden switch of the nonlinearity from repulsive to attractive. Then, we study oscillations of a single soliton in this setting, which models a recently conducted experiment in BEC. Basic characteristics of the MI-generated array, such as the number of solitons and their mobility, are reported as functions of the strength and correlation length of the disorder, and of the total norm. For the single oscillating soliton, its survival rate is found. Main features of these dependences are explained qualitatively.

Flowing water in river, transported gas or oil in pipe, electric current in wire, moving goods on conveyor, molecular motors in living cell, and driving vehicles on a highway are various kinds of flow from physical or non-physical systems, yet each exhibits distinct characteristics. One of the attributes that distinguishes vehicular traffic flow from other flows as a special kind of 'fluid' is the so-called fundamental diagram - the relationships among traffic flow characteristics (e.g. flow, speed, and density) which are typically represented graphically. The fundamental diagram plays an essential role in transportation analysis and operations. For example, the study on traffic flow dynamics relies on input from the speed-density relationship to understand how a perturbation propagates among vehicles; a highway capacity analysis utilizes the speed-flow relationship to determine the level of service that the highway provides. Hence, sound mathematical models that better represent these relationships build a solid foundation for traffic flow analysis and efficient traffic control. Such an observation has motivated a variety of speed-density models since the path-breaking attempt by Greenshields in 1935. Departing from the Greenshields model, a number of models are proposed with varying degrees of success. Note that these models are in a single-equation form (single-regime model). Further improvements are made by decomposing the speed-density relationship into multiple pieces for better fitting. However, no matter single- or multi-regime, these models are deterministic in nature which essentially describes average system behaviors from a statistical perspective. The aim of this research, rather than to resolve the current controversies regarding sources of randomness, is to advance the modeling effort of the speed-density relationship from the deterministic domain to stochastic while still achieving mathematical elegance and empirical accuracy. To fulfill this end, a conceptual model was formulated assuming that speed is a random process of density and a random variable.

The fundamental diagram, as the graphical representation of the relationships among traffic flow, speed, and density, has been the foundation of traffic flow theory and transportation engineering. Seventy-five years after the seminal Greenshields model, a variety of models have been proposed to mathematically represent the speed–density relationship which underlies the fundamental diagram. Observed in these models was a clear path toward two competing goals: mathematical elegance and empirical accuracy. As the latest development of such a pursuit, this paper presents a family of speed–density models with varying numbers of parameters. All of these models perform satisfactorily and have physically meaningful parameters. In addition, speed variation with traffic density is accounted for; this enables statistical approaches to traffic flow analysis. The results of this paper not only improve our understanding of traffic flow but also provide a sound basis for transportation engineering studies.

The LWR model is of interest since it is simple and can successfully reproduce some essential features of traffic flow, such as the formation and propagation of traffic disturbances. In this article, we investigate the LWR model from an uncertainty perspective. We attempt to analyse how reliable the LWR model prediction will be if the fundamental diagram (FD) in use is not accurately specified. To fulfil this end, we postulate a flux function (equivalently a FD) driven by a random free flow speed, which accommodates the uncertain feature observed in the speed–density data. We provide essential mathematical properties and solution schemes of the LWR model with the probabilistic FD. In case studies, the approach to evaluate the uncertainty of traffic disturbance propagation with this model is presented. We find that if FD in a LWR model cannot be perfectly specified, the uncertainty associated with the location of a traffic disturbance would increase over time. In contrast, the magnitude of the traffic disturbance can still be accurately predicted.

The analogy of traffic flow to water flow has been well known. But some essential differences exist, e.g. anisotropy, viscosity, and the extent to which conservation law holds. Not surprisingly, this is due to different manners in which water particles and vehicles move and interact. In this study, we investigate and model the viscous behavior exhibited by traffic flow. We attribute the viscous effect of traffic flow to drivers' heterogeneity in terms of their preferred driving speeds. A model incorporating the diffusion effect is developed based on the idea of characteristic curve. In particular, the governing equation of the traveling platoon is explicitly derived from this model. In accordance with our postulation, the derived equation includes one viscosity term. The proposed model is best suited for the analysis of the local fine structure of traffic flow which is conventionally represented as a shock.

1 The fundamental diagram, as the graphical representation of the relationships 2 among traffic flow, speed, and density, has been the foundation of traffic flow the3 ory and transportation engineering. 75 years after the seminal Greenshields model, a 4 variety of models have been proposed to mathematically represent the speed-density 5 relationship which underlies the fundamental diagram. Observed in these models was 6 a clear path toward two competing goals: mathematical elegance and empirical ac7 curacy. As the latest development of such a pursuit, this paper presents a family of 8 speed-density models with varying numbers of parameters. All of these models perform 9 satisfactorily and have physically meaningful parameters. In addition, speed variation 10 with traffic density is accounted for; this enables statistical approaches to traffic flow 11 analysis. The results of this paper not only improve our understanding of traffic flow 12 but also provide a sound basis for transportation engineering studies. 13 TRB 2010 Annual Meeting CD-ROM Paper revised from original submittal. H. Wang, J. Li, Q-Y. Chen and D. Ni 3

We consider effects of random time modulation of the nonlinearity coefficient on the dynamics of one- and two-dimensional (1D and 2D) solitary waves in the nonlinear Schrödinger equation (NLSE). In particular, the cases of a single Gaussian random variable, and a temporally correlated Gaussian process are considered. In the 1D case, we demonstrate the robustness of solitons against the random nonlinearity management. In the 2D case, the share (percentage) of realizations that lead to collapse of a localized pulse is computed, in order to quantify the effect of the randomness in preventing the collapse. Dependences of this share on the mean value, standard deviation, and correlation length of the random process are obtained, and, whenever possible, compared to analytical predictions.

Traffic flow is a many-car system with complex and stochastic movement. It is difficult to describe the system dynamics solely by using deterministic tools. Therefore, a stochastic speed-density relationship is proposed here as a further step forward to overcome the well known drawbacks of deterministic models. Modeling results show that by taking care of second order statistics (i.e., mean and variance) a stochastic speed-density model is suitable for describing the observed phenomenon as well as matching the empirical data. Starting from here, a stochastic fundamental diagram of traffic flow could be established. The stochastic speed-density relationship model can potentially be used for real-time on-line prediction and to explain phenomenons in a similar manner.

The randomness is prevalent in real life. In particular, it accounts for the appearance of some especially interested phenomena in trans- portation system, say the spontaneous congestion. For this reason, we postulate the LWR model with stochastic setting, essentially through defining a random flux function driven by random free flow speed. Its connection to the classical LWR model is briefly discussed and proper- ties essential for developing a flux-splitting scheme are given. A third order ENO-FD (essentially non-oscillatory finite difference) algorithm is devised and implemented to solve the postulated model numeri- cally. We give one example based on the given model, illustrating the evaluation of predictability of traffic flow disturbance propagation in uncertain circumstance.

We derive a novel finite volume method for the elliptic equation, using the framework of mixed finite element methods to discretize the pressure and velocities on two different grids (covolumes), triangular (tetrahedral) mesh and control volume mesh. The new discretization is defined for tensor diffusion coefficient and well suited for heterogeneous media. When the control volumes are created by connecting the center of gravity of each triangle to the midpoints of its edges, we show that the discretization is stable and first order accurate for both scalar and vector unknowns. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Abstract Multi-point flux approximations (MPFAs) have been developed in recent years to improve the accuracy of the flux computation for reservoir simulation when the reservoir grids are not K-orthogonal. Recent studies, however, have shown that commonly used MPFA approaches could pose severe limitations. Specifically, it has been observed that MPFA can cause severe nonphysical oscillations in numerical solutions for the single-phase pressure equation if the permeability field is strongly anisotropic. Some researchers have linked those numerical oscillations to the lack of monotonicity in the solution matrix for the discretization problem and efforts have been made to improve MPFA so that matrix monotonicity can be restored. This paper will present new techniques, collectively called enriched multi-point flux approximations (EMPFAs), to reduce or eliminate numerical oscillations in the solutions. The key to improving the consistency and accuracy of the local flux calculations is the introduction of new temporary unknowns and the application of better pressure interpolation techniques. EMPFA can be used for modeling general multiphase flows in 2-dimensional or 3-dimensional reservoir models, and the grid can be Cartesian, Voronoi, or even arbitrary. Simulation results will demonstrate the differences among EMPFA, MPFA, and the original two-point flux approximation (TPFA).

It is well known that the two-point flux approximation, a numerical scheme used in most commercial reservoir simulators, has O(1) error when grids are not K-orthogonal. In the last decade, the multi-point flux approximations have been developed as a remedy. However, non-physical oscillations can appear when the anisotropy is really strong. We found out the oscillations are closely related to the poor approximation of pressure gradient in the flux computation.In this paper, we propose the control volume enriched multi-point flux approximation (EMPFA) for general diffusion problems on polygonal and polyhedral meshes. Non-physical oscillations are not observed for realistic and strongly anisotropic heterogeneous material properties described by a full tensor. Exact linear solutions are recovered for grids with non-planar interfaces, and a first and second order convergence are achieved for the flux and scalar unknowns, respectively.

In this paper, we present several systematic techniques, based on the Voronoi diagram and its variants, to partition a one- and two-dimensional simplex. The Fekete points are used as input to generate the Voronoi diagram, as they concentrate near the edges and are almost optimal for polynomial interpolation in a simplex.Spectral (finite) volume reconstructions on the resultant partitions have small Lebesgue constants. When using the Dubiner basis, the reconstruction matrix is well conditioned. Moreover, the total number of edges of the partitions ( the total work when being used in spectral volume methods) is shown to be at most twice the minimum number of edges of all partitions for reconstructions of the same order accuracy. These suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages.

We examine the merits of using prolate spheroidal wave functions (PSWFs) as basis functions when solving hyperbolic PDEs using pseudospectral methods.The relevant approximation theory is reviewed and some new approximation results in Sobolev spaces are established. An optimal choice of the band-limit parameter for PSWFs is derived for single-mode functions.Our conclusion is that one might gain from using the PSWFs over the traditional Chebyshev or Legendre methods in terms of accuracy and efficiency for marginally resolved broadband solutions.