Michael Herty

Gas flow in pipeline networks

We introduce a model for gas flow in pipeline networks based on the isothermal Euler equations. We model the intersection of multiple pipes by posing an additional assumption on the pressure at the interface. We give a method to obtain solutions to the gas network problem and present numerical results for sample networks.

Coupling conditions for gas networks governed by the isothermal Euler equations

We investigate coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations. We discuss intersections of pipes by considering solutions to Riemann problems. We introduce additional assumptions to obtain a solution near the intersection and we present numerical results for sample networks.

Modeling, Simulation, and Optimization of Traffic Flow Networks

A new model for highway traffic networks based on a detailed description of the junctions is presented. To obtain suitable conditions at the junctions, multilane equations are introduced and investigated. The new model is compared with currently known models for traffic flow networks for several situations. Finally, the model is used for network simulation and optimization.

Coupling Conditions for a Class of Second-Order Models for Traffic Flow

This paper deals with a model for traffic flow based on a system of conservation laws [A. Aw and M. Rascle, SIAM J. Appl. Math., 60 (2000), pp. 916-938]. We construct a solution of the Riemann problem at an arbitrary junction of a road network. Our construction provides a solution of the full system. In particular, all moments are conserved.

Existence of Solutions for Supply Chain Models Based on Partial Differential Equations

We consider a model for supply chains governed by partial differential equations. The mathematical properties of a continuous model are discussed and existence and uniqueness are proven. Moreover, Lipschitz continuous dependence on the initial data is proven. We make use of the front tracking method to construct approximate solutions. The obtained results extend the preliminary work of [S. Gottlich, M. Herty, and A. Klar, Commun. Math. Sci., 3 (2005), pp. 545–559].

Flows on networks: recent results and 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.

Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach

This book offers a state-of-the-art introduction to the mathematical theory of supply chain networks, focusing on supply chain networks described by partial differential equations (PDEs). The authors discuss modeling of complex supply networks as well as their mathematical theory; explore modeling, simulation, and optimization of some of the discussed models; and present analytical and numerical results on optimization problems. Real-world examples are given to demonstrate the applicability of the presented approaches. Audience: Graduate students and researchers who are interested in the theory of supply chain networks described by PDEs will find this book useful. It can also be used in advanced graduate-level courses on modeling of physical phenomena, as well as introductory courses on supply chain theory. Contents: Preface; Chapter 1: Introduction; Chapter 2: Mathematical Preliminaries Chapter 3: Basic Queueing Models; Chapter 4: Models Based on Ordinary Differential Equations; Chapter 5: Models Based on Partial Differential Equations; Chapter 6: Continuum-Discrete Models; Chapter 7: Control and Optimization Problem for Networks; Chapter 8: Computational Results; Bibliography; Index

Gas Pipeline Models Revisited: Model Hierarchies, Nonisothermal Models, and Simulations of Networks

By using asymptotic analysis we derive most of the known and also new nonisothermal pipeline models starting from transient gas equations. We introduce proper scalings to identify valid regimes for the derived models and extend them to networks. Finally, we perform numerical simulations on a single pipe as well-posed as on a small network. We compare both isothermal and nonisothermal flow and pressure predictions with results obtained from the literature.

Control of Continuum Models of Production Systems

A production system which produces a large number of items in many steps can be modelled as a continuous flow problem. The resulting hyperbolic partial differential equation (PDE) typically is nonlinear and nonlocal, modeling a factory whose cycle time depends nonlinearly on the work in progress. One of the few ways to influence the output of such a factory is by adjusting the start rate in a time dependent manner. We study two prototypical control problems for this case: (i) demand tracking where we determine the start rate that generates an output rate which optimally tracks a given time dependent demand rate and (ii) backlog tracking which optimally tracks the cumulative demand. The method is based on the formal adjoint method for constrained optimization, incorporating the hyperbolic PDE as a constraint of a nonlinear optimization problem. We show numerical results on optimal start rate profiles for steps in the demand rate and for periodically varying demand rates and discuss the influence of the nonlinearity of the cycle time on the limits of the reactivity of the production system. Differences between perishable and non-perishable demand (demand versus backlog tracking) are highlighted.

Optimization criteria for modelling intersections of vehicular traffic flow

We consider coupling conditions for the “Aw–Rascle” (AR) traffic flow model at an arbitrary road intersection. In contrast with coupling conditions previously introduced in [10] and [7], all the moments of the AR system are conserved and the total flux at the junction is maximized. This nonlinear optimization problem is solved completely. We show how the two simple cases of merging and diverging junctions can be extended to more complex junctions, like roundabouts. Finally, we present some numerical results.