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 ...

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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

We consider a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic equation on a network. Coupling conditions for the macroscopic equations are derived from the kinetic coupling conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the combination of kinetic half-space problems with Riemann problems at the junction. Detailed numerical comparisons between the different models show the agreement of the coupling conditions for the case of tripod networks.

Kinetic layers and coupling conditions for nonlinear scalar equations on networks

This review focuses on kinetic and macroscopic models for the migration of cells in fiber structures. Typical applications of cell migration models in such geometries are tumor cell invasion into tissue, or tissue-engineering and the movement of fibroblasts on artificial scaffolds during wound healing.

Kinetic and Moment Models for Cell Motion in Fiber Structures

In this paper we construct a numerical solver for the Saint Venant equations. Special attention is given to the balancing of the source terms, including the bottom slope and variable crosssectional profiles. Therefore a special discretization of the pressure law is used, in order to transfer analytical properties to the numerical method. Based on this approximation a wellbalanced solver is developed, assuring the C-property and depth positivity. The performance of this method is studied in several test cases focusing on accurate capturing of steady states.

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A well-balanced solver for the Saint Venant equations with variable cross-section

In this paper we consider kinetic and associated macroscopic models for chemotaxis on a network. Coupling conditions at the nodes of the network for the kinetic problem are presented and used to derive coupling conditions for the macroscopic approximations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations asymptotic preserving relaxation schemes are extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

Kinetic and related macroscopic models for chemotaxis on networks

In this article, we present a numerical solver for simulating district heating networks. The method applies a local time stepping to networks of linear advection equations. Numerical diffusion as well as the computational effort on each edge is reduced significantly. The combination with high order coupling and reconstruction techniques leads to a very efficient scheme.

/pdf/local-time-stepping-method-for-district-heating-networks-4fbcdldb27.pdf

Raul Borsche

Papers

Kinetic layers and coupling conditions for nonlinear scalar equations on networks

Kinetic and Moment Models for Cell Motion in Fiber Structures

A well-balanced solver for the Saint Venant equations with variable cross-section

Kinetic and related macroscopic models for chemotaxis on networks

Local Time Stepping Method for District Heating Networks