# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

September 2009 , Volume 4 , Issue 3

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2009, 4(3): 431-451 doi: 10.3934/nhm.2009.4.431 +[Abstract](628) +[PDF](2014.8KB)
Abstract:
We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics, proposed in [ A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66 (2006), pp. 921--944]. This model takes into account interactions of every vehicle with other vehicles ahead ("look-ahead'' rule) and can be written as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the non-oscillatory central schemes, which belong to a class of Godunov-type projection-evolution methods. In this framework, a solution, computed at a certain time, is first approximated by a piecewise polynomial function, which is then evolved to the next time level according to the integral form of the conservation law. Most Godunov-type schemes are based on upwinding, which requires solving (generalized) Riemann problems. However, no (approximate) Riemann problem solver is available for conservation laws with global fluxes. Therefore, central schemes, which are Riemann-problem-solver-free, are especially attractive for the studied traffic flow model. Our numerical experiments demonstrate high resolution, stability, and robustness of the proposed methods, which are used to numerically investigate both dispersive and smoothing effects of the global flux.
We also modify the model by Sopasakis and Katsoulakis by introducing a more realistic, linear interaction potential that takes into account the fact that a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The central schemes are extended to the modified model. Our numerical studies clearly suggest that in the case of a good visibility, the new model yields solutions that seem to better correspond to reality.
2009, 4(3): 453-468 doi: 10.3934/nhm.2009.4.453 +[Abstract](557) +[PDF](249.2KB)
Abstract:
We consider the continuous Laplacian on infinite locally finite networks under natural transition conditions as continuity at the ramification nodes and Kirchhoff flow conditions at all vertices. It is well known that one cannot reconstruct the shape of a finite network by means of the eigenvalues of the Laplacian on it. The same is shown to hold for infinite graphs in a $L^\infty$-setting. Moreover, the occurrence of eigenvalue multiplicities with eigenspaces containing subspaces isomorphic to $\l^\infty(\ZZ)$ is investigated, in particular in trees and periodic graphs.
2009, 4(3): 469-500 doi: 10.3934/nhm.2009.4.469 +[Abstract](601) +[PDF](1239.2KB)
Abstract:
The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneous ventilation process. To that end, we treat the bronchial tree as an infinite resistive tree, we endow the space of pressures at bifurcating nodes with the natural energy norm (rate of dissipated energy), and we characterise the pressure field at its boundary (i.e. set of simple paths to infinity). In a second step, we embed the infinite collection of leafs in a bounded domain Ω$\subset \RR^d$, and we establish some regularity properties for the corresponding pressure field. In particular, for the infinite counterpart of a regular, healthy lung, we show that the pressure field lies in a Sobolev space $H^{s}$(Ω), with $s \approx 0.45$. This allows us to propose a model for the ventilation process that takes the form of a boundary problem, where the role of the boundary is played by a full domain in the physical space, and the elliptic operator is defined over an infinite dyadic tree.
2009, 4(3): 501-526 doi: 10.3934/nhm.2009.4.501 +[Abstract](615) +[PDF](307.6KB)
Abstract:
Boltzmann Maps are a class of discrete dynamical systems that may be used in the study of complex chemical reaction processes. In this paper they are generalized to open systems allowing the description of non-stoichiometrically balanced reactions with unequal reaction rates. We show that they can be widely used to describe the relevant dynamics, leading to interesting insights on the multi-stability problem in networks of chemical reactions. Necessary conditions for multistability are thus identified. Our findings indicate that the dynamics produced by laws like the mass action law, can hardly produce multistable phenomena. In particular, we prove that they cannot do it in a wide range of chemical reactions.
2009, 4(3): 527-536 doi: 10.3934/nhm.2009.4.527 +[Abstract](568) +[PDF](174.6KB)
Abstract:
Critical threshold phenomena in a one dimensional quasi-linear hyperbolic model of blood flow with viscous damping are investigated. We prove global in time regularity and finite time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the blood flow model. New results are obtained showing that the class of data that leads to global smooth solutions includes the data with negative initial Riemann invariant slopes and that the magnitude of the negative slope is not necessarily small, but it is determined by the magnitude of the viscous damping. For the data that leads to shock formation, we show that shock formation is delayed due to viscous damping.
2009, 4(3): 537-575 doi: 10.3934/nhm.2009.4.537 +[Abstract](651) +[PDF](383.6KB)
Abstract:
The topic of security often enters in many real world situations. In this paper we focus on security of networks on which it is based the delivery of services and goods (e.g. water and electric supply networks) the transfer of data (e.g. web and telecommunication networks), the movement of transport means (e.g. road networks), etc... We use a fluid dynamic framework, networks are described by nodes and lines and our analysis starts from an equilibrium status: the flows are constant in time and along the lines. When a failure occurs in the network a shunt changes the topology of the network and the flows adapt to it reaching a new equilibrium status. The question we consider is the following: is the new equilibrium satisfactory in terms of achieved quality standards? We essentially individuate, for regular square networks, the nodes whose breakage compromises the quality of the flows. It comes out that networks which allow circular flows are the most robust with respect to damages.
2009, 4(3): 577-604 doi: 10.3934/nhm.2009.4.577 +[Abstract](532) +[PDF](338.6KB)
Abstract:
Thin periodic structures depend on two interrelated small geometric parameters $\varepsilon$ and $h(\varepsilon)$ which control the thickness of constituents and the cell of periodicity. We study homogenisation of elasticity theory problems on these structures by method of asymptotic expansions. A particular attention is paid to the case of critical thickness when $\lim_{\varepsilon\to 0} h(\varepsilon)\varepsilon^{-1}$ is a positive constant. Planar grids are taken as a model example.
2009, 4(3): 605-623 doi: 10.3934/nhm.2009.4.605 +[Abstract](611) +[PDF](2614.8KB)
Abstract:
Starting from a continuous congested traffic framework recently introduced in [8], we present a consistent numerical scheme to compute equilibrium metrics. We show that equilibrium metric is the solution of a variational problem involving geodesic distances. Our discretization scheme is based on the Fast Marching Method. Convergence is proved via a $\Gamma$-convergence result and numerical results are given.

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