1,721,061 research outputs found
Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows
No full textWe review the recent results and present new ones on a deterministic follow-the-leader particle approximation of first-and second-order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first-order LighthillâWhithamâRichards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretized vehicles seen as moving particles. The convergence of the scheme in the many particle limit toward the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the initialâboundary value problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case, we prove that our scheme is convergent strongly in L1up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second-order AwâRascleâZhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases
Follow-the-Leader Approximations of Macroscopic Models for Vehicular and Pedestrian Flows
No full textWe review recent results and present new ones on a deterministic follow-the-leader particle approximation of first and second order models for traffic flow and pedestrian movements. We start by constructing the particle scheme for the first order Lighthill-Whitham-Richards (LWR) model for traffic flow. The approximation is performed by a set of ODEs following the position of discretised vehicles seen as moving particles. The convergence of the scheme in the many particle limit towards the unique entropy solution of the LWR equation is proven in the case of the Cauchy problem on the real line. We then extend our approach to the Initial-Boundary Value Problem (IBVP) with time-varying Dirichlet data on a bounded interval. In this case we prove that our scheme is convergent strongly in L1 up to a subsequence. We then review extensions of this approach to the Hughes model for pedestrian movements and to the second order Aw-Rascle-Zhang (ARZ) model for vehicular traffic. Finally, we complement our results with numerical simulations. In particular, the simulations performed on the IBVP and the ARZ model suggest the consistency of the corresponding schemes, which is easy to prove rigorously in some simple cases
Stable and convergent discontinuous galerkin methods for hyperbolic and viscous systems of conservation laws
No full textDespite the classical well-posedness theorem for entropy weak solutions of scalar conservation laws, some theoretical and numerical evidence cast doubt on the appropriateness of this solution paradigm for multidimensional hyperbolic systems. It has been conjectured that the more general entropy measure-valued (EMV) solutions ought to be considered as the adequate notion of the solution. Building on previous results, we prove that bounded solutions of a certain class of space-time discontinuous Galerkin (DG) schemes converge to an EMV solution. The novelty in our work is that no streamline-diffusion terms are used for stabilization, in contrary to the main role of such stabilizations in the existing analysis of DG schemes. Our approach conforms to the way DG schemes were originally proposed, and are most often used in practiceIn the case of scalar problems, this result is strengthened to obtain the convergence to the entropy weak solution, via the proof of -boundedness of the solution as well as its consistency with all entropy inequalities. As a main step in the boundedness proof, we show the coercivity of the shock-capturing operator employing new arguments from polynomial inequalities. For viscous conservation laws, we extend our framework to general convection-diffusion systems, with both nonlinear convection and nonlinear diffusion, such that the entropy stability of the scheme is preserved. Starting from a mixed formulation, we handle the difficulties arising from the nonlinearity of the viscous flux by an additional projection. We prove the entropy stability of the corresponding primal form for different treatments of the viscous flux; thus unifying the existing results in the literature as well as establishing the entropy stability for less-analyzed methods. Our analysis is also valid for the case of degenerate diffusion. Considering quasilinear elliptic problems in scalar settings, we prove that the proposed approach for viscous discretization is asymptotically consistent and adjoint consistent. For the special case of strongly monotone and globally Lipschitz problems, we prove the uniqueness and stability of the numerical solution. For this class of operators, we also provethe optimal convergence to the exact solution with respect to mesh size, in both energy and norms. Such optimal convergence rates for asymptotically (adjoint) consistent schemes have been observed before in numerical experiments
Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
Full text linkLet u(x,t) be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose u sub epsilon(x,t) is the solution of an approximate viscosity regularization, where epsilon greater than 0 is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation u sub epsilon, pointwise values of u and its derivatives can be recovered with an error as close to epsilon as desired. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport with discontinuous coefficients. The novelty of this approach is to use a (generalized) E-condition of the forward problem in order to deduce a W(exp 1,infinity) energy estimate for the discontinuous backward transport equation; this, in turn, leads one to an epsilon-uniform estimate on moments of the error u(sub epsilon) - u. This approach does not follow the characteristics and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Entropy functions for symmetric systems of conservation laws
No full textAbstractUsing a simple symmetrizability criterion, we show that symmetric systems of conservation laws are equipped with a one-parameter family of entropy functions
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