Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

We studied the Brownian motion of isolated ellipsoidal particles in water confined to two dimensions and elucidated the effects of coupling between rotational and translational motion. By using digital video microscopy, we quantified the crossover from short-time anisotropic to long-time isotropic diffusion and directly measured probability distributions functions for displacements. We confirmed and interpreted our measurements by using Langevin theory and numerical simulations. Our theory and observations provide insights into fundamental diffusive processes, which are potentially useful for understanding transport in membranes and for understanding the motions of anisotropic macromolecules.

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Brownian Motion of an ellipsoid

This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of ...

/pdf/vehicular-traffic-crowds-and-swarms-from-kinetic-theory-and-2e3l5tqiwc.pdf

Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives

The broad research thematic of flows on networks was addressed in recent years by many researchers, in the area of applied mathematics, with new models based on partial differential equations. The latter brought a significant innovation in a field previously dominated by more classical techniques from discrete mathematics or methods based on ordinary differential equations. In particular, a number of results, mainly dealing with vehicular traffic, supply chains and data networks, were collected in two monographs: Traffic flow on networks, AIMSciences, Springfield, 2006, and Modeling, simulation, and optimization of supply chains, SIAM, Philadelphia, 2010. The field continues to flourish and a considerable number of papers devoted to the subject is published every year, also because of the wide and increasing range of applications: from blood flow to air traffic management. The aim of the present survey paper is to provide a view on a large number of themes, results and applications related to this broad research direction. The authors cover different expertise (modeling, analysis, numeric, optimization and other) so to provide an overview as extensive as possible. The focus is mainly on developments which appeared subsequently to the publication of the aforementioned books.

https://www.ems-ph.org/journals/show_pdf.php?issn=2308-2151&vol=1&iss=1&rank=2

Flows on networks: recent results and perspectives

In this paper we present an approach to approximate numerically the solution of coupled hyperbolic conservation laws. The coupling is achieved through a fixed interface, in which interface conditions are linking the traces of both sides. The numerical solver is based on central methods, like the Rusanov scheme, and does not use the structure of the Riemann Problem. It consists in balancing the effects of the waves that enter the interface. The scheme is well balanced with respect to all the piecewise constant equilibria associated with the interface condition and is able to maintain exactly conservation properties of the interface conditions. A detailed analysis and several numerical tests show the quality of the method. Different applications, including sonic and transsonic flows and a multiphysic model are studied.

Numerical Approximation of Hyperbolic Systems Containing an Interface

We study the application of a recently introduced hierarchical description of traffic flow control by driver-assist vehicles to include lane changing dynamics. Lane-dependent feedback control strategies are implemented at the level of vehicles and the aggregate trends are studied by means of Boltzmann-type equations determining three different hydrodynamics based on the lane switching frequency. System of first order macroscopic equations describing the evolution of densities along the lanes are then consistently determined through a suitable closured strategy. Numerical examples are then presented to illustrate the features of the proposed hierarchical approach.

Kinetic-controlled hydrodynamics for multilane traffic models

Nonlocal conservation laws are used to describe various realistic instances of crowd behaviors. First, a basic analytic framework is established through an "ad hoc" well posedness theorem for systems of nonlocal conservation laws in several space dimensions interacting non locally with a system of ODEs. Numerical integrations show possible applications to the interaction of different groups of pedestrians, and also with other "agents".

Differential Equations Modeling Crowd Interactions

We consider an interacting system of one-dimensional structures modeling fibers with fiber-fiber interaction in a fiber lay-down process. The resulting microscopic system is investigated by looking at different asymptotic limits of the corresponding stochastic model. Equations arising from mean-field and diffusion limits are considered. Furthermore, numerical methods for the stochastic system and its mean-field counterpart are discussed. A numerical comparison of solutions corresponding to the different scales (microscopic, mesoscopic, and macroscopic) is included.

A retarded mean-field approach for interacting fiber structures

Kinetic models for vehicular traffic are reviewed and considered from the point of view of deriving macroscopic equations. A derivation of the associated macroscopic traffic flow equations leads to different types of equations: in certain situations modified Aw-Rascle equations are obtained. On the other hand, for several choices of kinetic parameters new Hamilton-Jacobi type traffic equations are found. Associated microscopic models are discussed and numerical experiments are presented discussing several situations for highway traffic and comparing the different models.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2013/0011/0003/CMS-2013-0011-0003-a004.pdf

Raul Borsche

Papers

Numerical Approximation of Hyperbolic Systems Containing an Interface

Kinetic-controlled hydrodynamics for multilane traffic models

Differential Equations Modeling Crowd Interactions

A retarded mean-field approach for interacting fiber structures

Kinetic derivation of a Hamilton-Jacobi traffic flow model