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Networks & Heterogeneous Media

Open Access Articles

On a vorticity-based formulation for reaction-diffusion-Brinkman systems
Verónica Anaya, Mostafa Bendahmane, David Mora and Ricardo Ruiz Baier
2018, 13(1): 69-94 doi: 10.3934/nhm.2018004 +[Abstract](157) +[HTML](105) +[PDF](6157.76KB)

We are interested in modelling the interaction of bacteria and certain nutrient concentration within a porous medium admitting viscous flow. The governing equations in primal-mixed form consist of an advection-reaction-diffusion system representing the bacteria-chemical mass exchange, coupled to the Brinkman problem written in terms of fluid vorticity, velocity and pressure, and describing the flow patterns driven by an external source depending on the local distribution of the chemical species. A priori stability bounds are derived for the uncoupled problems, and the solvability of the full system is analysed using a fixed-point approach. We introduce a primal-mixed finite element method to numerically solve the model equations, employing a primal scheme with piecewise linear approximation of the reaction-diffusion unknowns, while the discrete flow problem uses a mixed approach based on Raviart-Thomas elements for velocity, Nédélec elements for vorticity, and piecewise constant pressure approximations. In particular, this choice produces exactly divergence-free velocity approximations. We establish existence of discrete solutions and show their convergence to the weak solution of the continuous coupled problem. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme.

Numerical approximation of a coagulation-fragmentation model for animal group size statistics
Pierre Degond and Maximilian Engel
2017, 12(2): 217-243 doi: 10.3934/nhm.2017009 +[Abstract](350) +[HTML](130) +[PDF](819.2KB)

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equilibrium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are validated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates.

Numerical network models and entropy principles for isothermal junction flow
Gunhild A. Reigstad
2014, 9(1): 65-95 doi: 10.3934/nhm.2014.9.65 +[Abstract](276) +[PDF](1803.6KB)
We numerically explore network models which are derived for the isothermal Euler equations. Previously we proved the existence and uniqueness of solutions to the generalized Riemann problem at a junction under the conditions of monotone momentum related coupling constant and equal cross-sectional areas for all connected pipe sections. In the present paper we extend this proof to the case of pipe sections of different cross-sectional areas.
    We describe a numerical implementation of the network models, where the flow in each pipe section is calculated using a classical high-resolution Roe scheme. We propose a numerical treatment of the boundary conditions at the pipe-junction interface, consistent with the coupling conditions. In particular, mass is exactly conserved across the junction.
    Numerical results are provided for two different network configurations and for three different network models. Mechanical energy considerations are applied in order to evaluate the results in terms of physical soundness. Analytical predictions for junctions connecting three pipe sections are verified for both network configurations. Long term behaviour of physical and unphysical solutions are presented and compared, and the impact of having pipes with different cross-sectional area is shown.
Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models
Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov and Rodolfo R. Rosales
2013, 8(3): 745-772 doi: 10.3934/nhm.2013.8.745 +[Abstract](264) +[PDF](1439.2KB)
Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points
Jean-Yves Le Boudec
2013, 8(2): 529-540 doi: 10.3934/nhm.2013.8.529 +[Abstract](119) +[PDF](208.1KB)
Assume that a stochastic process can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.
Henri Berestycki, Danielle Hilhorst, Frank Merle, Masayasu Mimura and Khashayar Pakdaman
2013, 8(1): i-iii doi: 10.3934/nhm.2013.8.1i +[Abstract](113) +[PDF](93.9KB)
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He is a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

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Fabio Camilli, Italo Capuzzo Dolcetta and Maurizio Falcone
2012, 7(2): i-ii doi: 10.3934/nhm.2012.7.2i +[Abstract](156) +[PDF](101.9KB)
The theory of Mean Field Games (MFG, in short) is a branch of the theory of Differential Games which aims at modeling and analyzing complex decision processes involving a large number of indistinguishable rational agents who have individually a very small influence on the overall system and are, on the other hand, influenced by the mass of the other agents. The name comes from particle physics where it is common to consider interactions among particles as an external mean field which influences the particles. In spite of the optimization made by rational agents, playing the role of particles in such models, appropriate mean field equations can be derived to replace the many particles interactions by a single problem with an appropriately chosen external mean field which takes into account the global behavior of the individuals.

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Liquidity generated by heterogeneous beliefs and costly estimations
Min Shen and Gabriel Turinici
2012, 7(2): 349-361 doi: 10.3934/nhm.2012.7.349 +[Abstract](224) +[PDF](359.4KB)
We study the liquidity, defined as the size of the trading volume, in a situation where an infinite number of agents with heterogeneous beliefs reach a trade-off between the cost of a precise estimation (variable depending on the agent) and the expected wealth from trading. The "true" asset price is not known and the market price is set at a level that clears the market. We show that, under some technical assumptions, the model has natural properties such as monotony of supply and demand functions with respect to the price, existence of an equilibrium and monotony with respect to the marginal cost of information. We also situate our approach within the Mean Field Games (MFG) framework of Lions and Lasry which allows to obtain an interpretation as a limit of Nash equilibrium for an infinite number of agents.
Recognition of crowd behavior from mobile sensors with pattern analysis and graph clustering methods
Daniel Roggen, Martin Wirz, Gerhard Tröster and Dirk Helbing
2011, 6(3): 521-544 doi: 10.3934/nhm.2011.6.521 +[Abstract](395) +[PDF](5425.0KB)
Mobile on-body sensing has distinct advantages for the analysis and understanding of crowd dynamics: sensing is not geographically restricted to a specific instrumented area, mobile phones offer on-body sensing and they are already deployed on a large scale, and the rich sets of sensors they contain allows one to characterize the behavior of users through pattern recognition techniques.
    In this paper we present a methodological framework for the machine recognition of crowd behavior from on-body sensors, such as those in mobile phones. The recognition of crowd behaviors opens the way to the acquisition of large-scale datasets for the analysis and understanding of crowd dynamics. It has also practical safety applications by providing improved crowd situational awareness in cases of emergency.
    The framework comprises: behavioral recognition with the user's mobile device, pairwise analyses of the activity relatedness of two users, and graph clustering in order to uncover globally, which users participate in a given crowd behavior. We illustrate this framework for the identification of groups of persons walking, using empirically collected data.
    We discuss the challenges and research avenues for theoretical and applied mathematics arising from the mobile sensing of crowd behaviors.
Rinaldo M. Colombo and Benedetto Piccoli
2011, 6(3): i-iii doi: 10.3934/nhm.2011.6.3i +[Abstract](110) +[PDF](158.2KB)
The research about crowd dynamics has undergone a dramatic development in the recent years. This fast advancement made it rather difficult for researchers in applied mathematics to keep contacts with the variety of analytical and numerical techniques recently introduced, as well as with the new problems being considered. Indeed, Crowd Dynamics is of interest to disciplines ranging from pure mathematical analysis to operation research, from numerical analysis to computer graphics, from model theory to statistics. The variety of MSC classifications labeling the papers of this special issue testifies the broadness of the subjects covered hereafter and, hence, also of this whole field.
   This special issue of Networks and Heterogeneous Media aims at bridging several different research directions of interest to applied mathematicians. Each of the present papers describes key problems of particular interest for the authors, points at the related most relevant techniques and includes the corresponding main results. The common spirit is to share, also with non specialists of the very same field, achieved results in their full depth.
Frédéric Coquel, Edwige Godlewski, Jean-Marc Hérard and Jacques Segré
2010, 5(3): i-ii doi: 10.3934/nhm.2010.5.3i +[Abstract](190) +[PDF](46.6KB)
Special Issue from the workshop New Trends in Model Coupling, Theory, Nu- merics and Applications (NTMC’09), Paris, September 2 − 4 2009.
This special issue comprises selected papers from the workshop New Trends in Model Coupling, Theory, Numerics and Applications (NTMC'09) which took place in Paris, September 2 - 4, 2009. The research of optimal technological solutions in a large amount of industrial systems requires to perform numerical simulations of complex phenomena which are often characterized by the coupling of models related to various space and/or time scales. Thus, the so-called multiscale modelling has been a thriving scientific activity which connects applied mathematics and other disciplines such as physics, chemistry, biology or even social sciences. To illustrate the variety of fields concerned by the natural occurrence of model coupling we may quote:
  • meteorology where it is required to take into account several turbulence scales or the interaction between oceans and atmosphere, but also regional models in a global description,
  • solid mechanics where a thorough understanding of complex phenomena such as propagation of cracks needs to couple various models from the atomistic level to the macroscopic level;
  • plasma physics for fusion energy for instance where dense plasmas and collisionless plasma coexist;
  • multiphase fluid dynamics when several types of flow corresponding to several types of models are present simultaneously in complex circuits;
  • social behaviour analysis with interaction between individual actions and collective behaviour.

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Georges Bastin, Alexandre M. Bayen, Ciro D'Apice, Xavier Litrico and Benedetto Piccoli
2009, 4(2): i-v doi: 10.3934/nhm.2009.4.2i +[Abstract](114) +[PDF](87.6KB)
1. Introduction: Management of canal networks at the age of information technology. With the miniaturization of sensors and their decreasing costs, the paradigm of instrumentation of the built infrastructure and the environment has now been underway for several years, leading to numerous successful and sometimes spectacular realizations such as the instrumentation of the Golden Gate with wire- less sensors a few years ago. The convergence of communication, control and sensing on numerous platforms including multi-media platforms has enabled engineers to augment physical infrastructure systems with an information layer, capable of real- time monitoring, with particular success in the health monitoring community. This paradigm has reached a level of maturity, revealed by the emergence of numerous technologies usable to monitor the built infrastructure. Supervisory Control And Data Acquisition (SCADA) systems are a perfect example of such infrastructure, which integrate sensing, communication and control. In the context of management of irrigation networks, the impact of this technology on the control of such systems has the potential of significantly improving efficiency of operations.

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Leonid Berlyand and V. V. Zhikov
2008, 3(3): i-ii doi: 10.3934/nhm.2008.3.3i +[Abstract](118) +[PDF](40.9KB)
This special issue contains selected papers on Homogenization Theory and related topics. It is dedicated to Eugene Khruslov on the occasion of his seventieth birthday. Professor Khruslov made pioneering contributions into this field.
Homogenization problems were first studied in the late nineteenth century (Poisson, Maxwell, Rayleigh) and early twentieth century (Einstein). These studies were based on deep physical intuition allowing these outstanding physicists to solve several specific important problems such as calculating the effective conductivity of a two-phase conductor and the effective viscosity of suspensions. It was not until the early 1960s that homogenization began to gain a rigorous mathematical footing which enabled it to be applied to a wide variety of problem in physics and mechanics. A number of mathematical tools such as the asymptotic analysis of PDEs, variational bounds, heterogeneous multiscale method, and the probabilistic techniques of averaging were developed. Although this theory is a well-established area of mathematics, many fascinating problems remain open. Interesting examples of such problems can be found in the papers of this issue.

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Dirk Helbing, Jürgen Jost and Alessandro Vespignani
2008, 3(2): i-ii doi: 10.3934/nhm.2008.3.2i +[Abstract](236) +[PDF](56.7KB)
This issue of Networks and Heterogeneous Media (NHM) collects selected papers of the European Conference on Complex Systems ’07, which took place in Dresden, Germany, from October 1 to 6, 2007.
Studying complex systems has enormously changed our view of the world. The discovery that actions and reactions are often disproportionate and that small per- turbations can cause tremendous responses, has led to new scientific disciplines such as catastrophe theory, chaos theory, and the theory of phase transitions.

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Alexandre M. Bayen, Naomi Ehrich Leonard and Benedetto Piccoli
2007, 2(4): i-ii doi: 10.3934/nhm.2007.2.4i +[Abstract](83) +[PDF](47.3KB)
This special issue was linked to a workshop, having the same title, held at the Centro De Giorgi of the Scuola Normale Superiore di Pisa on April 6th 2007 and centered around the broad theme: modelling and control of physical networks. The workshop was a one-day satellite event of Hybrid Systems: Computation and Control (HSCC 2007). We want to acknowledge sponsored projects for this workshop, in particular: Italian project PRIN2005 “Metodi di viscosit, metrici e di teoria del controllo in equazioni alle derivate parziali nonlineari”, coordinated by I. Capuzzo Dolcetta; INDAM Project 2005 “Traffic flows and optimization on complex networks”, coordinated by B. Piccoli (; Istituto per le Applicazioni del Calcolo “Mauro Picone” ( More information can be found at

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