Benedetto Piccoli | Rutgers at Camden (original) (raw)
Papers by Benedetto Piccoli
Axioms, Feb 7, 2021
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Control Problems for Conservation Laws with Traffic Applications
Conservation and/or balance laws on networks in the recent years have been the subject of intense... more Conservation and/or balance laws on networks in the recent years have been the subject of intense study, since a wide range of different applications in real life can be covered by such a research.
Control Problems for Conservation Laws with Traffic Applications
A vehicle with different (eventually controlled) dynamics from general traffic along a street may... more A vehicle with different (eventually controlled) dynamics from general traffic along a street may reduce the road capacity, thus generating a moving bottleneck, and can be used to act on the traffic flow. The interaction between the controlled vehicle and the surrounding traffic, and the consequent flow reduction at the bottleneck position, can be described either by a conservation law with space dependent flux function [200], or by a strongly coupled PDE-ODE system proposed in [112, 208].
Axioms, 2021
This paper uses empirical traffic data collected from three locations in Europe and the US to rev... more This paper uses empirical traffic data collected from three locations in Europe and the US to reveal a three-phase fundamental diagram with two phases located in the uncongested regime. Model-based clustering, hypothesis testing and regression analyses are applied to the speed–flow–occupancy relationship represented in the three-dimensional space to rigorously validate the three phases and identify their gaps. The finding is consistent across the aforementioned different geographical locations. Accordingly, we propose a three-phase macroscopic traffic flow model and a characterization of solutions to the Riemann problems. This work identifies critical structures in the fundamental diagram that are typically ignored in first- and higher-order models and could significantly impact travel time estimation on highways.
Journal of Differential Equations, 2020
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of... more Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for time-varying bottleneck speed, which corresponds to open-loop controls with bounded variation.
Communications in Mathematical Sciences, 2018
In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riem... more In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riemann solver (PRS) provides a solution at junctions by taking into consideration priorities for the incoming roads and maximization of through flux. We prove existence of solutions for the solver for junctions with up to two incoming and two outgoing roads and show numerically the comparison with previous Riemann solvers. Additionally, we introduce a second version of the solver that considers the priorities as softer constraints and illustrate numerically the differences between the two solvers.
Transportmetrica B: Transport Dynamics, 2015
We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks ... more We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model (Lighthill and Whitham, 1955; Richards, 1956). Derivation of the LKWM involves the variational principle for the Hamilton-Jacobi equation and junction models defined via the notions of demand and supply. We show that the proposed LKWM can be formulated as a system of differential algebraic equations (DAEs), which captures shock formation and propagation, as well as queue spillback. The DAE system, as we show in this paper, is the continuous-time counterpart of the link transmission model (Yperman et al., 2005). In addition, we present a solution existence theory for the continuous-time network model and investigate continuous dependence of the solution on the initial data, a property known as well-posedness. We test the DAE system extensively on several small and large networks and demonstrate its numerical efficiency.
Mathematical Models and Methods in Applied Sciences, 2017
We consider nonlinear transport equations with non-local velocity describing the time-evolution o... more We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic–Quinn procedure. Moreover, the control l...
Transportation Research Part B: Methodological, 2016
This paper establishes the continuity of the path delay operators for dynamic network loading (DN... more This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations (PDAEs). We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.
Automatica, 2017
For control-affine systems with a proper Lyapunov function, the classical procedure Jurdjevic-Qui... more For control-affine systems with a proper Lyapunov function, the classical procedure Jurdjevic-Quinn (see [21]) gives a well-known and widely used way of designing feedback controls that asymptotically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated at the same time. In this paper we give sufficient conditions under which this stabilization can be done by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. We thus obtain a sparse version of the classical Jurdjevic-Quinn theorem. We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying feedback, a sampled feedback, and a hybrid hysteresis. We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models.
Active Particles, Volume 1, 2017
In the present chapter we study the emergence of global patterns in large groups in first and sec... more In the present chapter we study the emergence of global patterns in large groups in first and second-order multi-agent systems, focusing on two ingredients that influence the dynamics: the interaction network and the state space. The state space determines the types of equilibrium that can be reached by the system. Meanwhile, convergence to specific equilibria depends on the connectivity of the interaction network and on the interaction potential. When the system does not satisfy the necessary conditions for convergence to the desired equilibrium, control can be exerted, both on finite-dimensional systems and on their mean-field limit.
Procedia - Social and Behavioral Sciences, 2012
Mathematical Control & Related Fields, 2013
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. B... more This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012. We focus on a controlled Cucker-Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues. We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which "sparse is better". We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
Mathematical Models and Methods in Applied Sciences, 2015
Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker–Smale (2007), t... more Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker–Smale (2007), there has been a lot of recent works on models of self-alignment and consensus dynamics. Self-organization has so far been the main driving concept of this research direction. However, the evidence that in practice self-organization does not necessarily occur (for instance, the achievement of unanimous consensus in government decisions) leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most "economical" way to achieve a certain outcome. Our paper precisely addressed the issue of finding the sparsest control strategy in order to lead us optimally towards a given outcome, in this case the achievement of a state where the group will be able...
Networks & Heterogeneous Media, 2006
We consider a mathematical model for fluid-dynamic flows on networks which is based on conservati... more We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occurr and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.
EMS Surveys in Mathematical Sciences, 2014
The broad research thematic of flows on networks was addressed in recent years by many researcher... more 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. 1 The author acknowledges partial support of 2013 GNAMPA project "Leggi di Conservazione: Teoria e Applicazioni". 2 The author acknowledges support by BMBF KinOpt, DFG Cluster of Excellence EXC128 and DAAD 54365630, 55866082. 3 The author acknowledges partial support of NSF Research Network in the Mathematical Sciences KI-Net "Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # : 1107444.
Contemporary Mathematics, 1999
In this paper we establish the existence of nonclassical entropy solutions for the Cauchy problem... more In this paper we establish the existence of nonclassical entropy solutions for the Cauchy problem associated with a conservation law having a nonconvex flux-function. Instead of the classical Oleinik entropy criterion, we use a single entropy inequality supplemented with a kinetic relation. We prove that these two conditions characterize a unique nonclassical Riemann solver. Then we apply the wave-front tracking method to the Cauchy problem. By introducing a new total variation functional, we can prove that the corresponding approximate solutions converge strongly to a nonclassical entropy solution.
SIAM Journal on Scientific Computing, 2011
Numerical techniques for the simulation of an ODE-PDE model for supply chains are presented. Firs... more Numerical techniques for the simulation of an ODE-PDE model for supply chains are presented. First we describe a scheme, based on Upwind and explicit Euler methods, provide corrections to maintain positivity of solutions, prove convergence and provide convergence rate. The latter is achieved via comparison with Wave Front Tracking solutions and the use of generalized tangent vectors. Different choice of time and space meshes give similar results, both for CPU times and numerical errors. Fast algorithms, based on an accurate choice of time and space meshes and data structures, are then proposed, achieving high computational gains.
SIAM Journal on Applied Mathematics, 2011
An extension of the Colombo phase transition model is proposed. The congestion phase is described... more An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann problem in the presence of phase transitions is obtained through the construction of a Riemann solver, which enables the definition of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell-Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed.
SIAM Journal on Applied Dynamical Systems, 2008
In this paper we introduce a computation algorithm to trace car paths on road networks, whose loa... more In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed by two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are showed. Moreover a convergence result for wave front tracking approximate solutions, with BV initial data on a single road is established.
Axioms, Feb 7, 2021
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Control Problems for Conservation Laws with Traffic Applications
Conservation and/or balance laws on networks in the recent years have been the subject of intense... more Conservation and/or balance laws on networks in the recent years have been the subject of intense study, since a wide range of different applications in real life can be covered by such a research.
Control Problems for Conservation Laws with Traffic Applications
A vehicle with different (eventually controlled) dynamics from general traffic along a street may... more A vehicle with different (eventually controlled) dynamics from general traffic along a street may reduce the road capacity, thus generating a moving bottleneck, and can be used to act on the traffic flow. The interaction between the controlled vehicle and the surrounding traffic, and the consequent flow reduction at the bottleneck position, can be described either by a conservation law with space dependent flux function [200], or by a strongly coupled PDE-ODE system proposed in [112, 208].
Axioms, 2021
This paper uses empirical traffic data collected from three locations in Europe and the US to rev... more This paper uses empirical traffic data collected from three locations in Europe and the US to reveal a three-phase fundamental diagram with two phases located in the uncongested regime. Model-based clustering, hypothesis testing and regression analyses are applied to the speed–flow–occupancy relationship represented in the three-dimensional space to rigorously validate the three phases and identify their gaps. The finding is consistent across the aforementioned different geographical locations. Accordingly, we propose a three-phase macroscopic traffic flow model and a characterization of solutions to the Riemann problems. This work identifies critical structures in the fundamental diagram that are typically ignored in first- and higher-order models and could significantly impact travel time estimation on highways.
Journal of Differential Equations, 2020
Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of... more Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for time-varying bottleneck speed, which corresponds to open-loop controls with bounded variation.
Communications in Mathematical Sciences, 2018
In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riem... more In this article we introduce a new Riemann solver for traffic flow on networks. The Priority Riemann solver (PRS) provides a solution at junctions by taking into consideration priorities for the incoming roads and maximization of through flux. We prove existence of solutions for the solver for junctions with up to two incoming and two outgoing roads and show numerically the comparison with previous Riemann solvers. Additionally, we introduce a second version of the solver that considers the priorities as softer constraints and illustrate numerically the differences between the two solvers.
Transportmetrica B: Transport Dynamics, 2015
We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks ... more We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model (Lighthill and Whitham, 1955; Richards, 1956). Derivation of the LKWM involves the variational principle for the Hamilton-Jacobi equation and junction models defined via the notions of demand and supply. We show that the proposed LKWM can be formulated as a system of differential algebraic equations (DAEs), which captures shock formation and propagation, as well as queue spillback. The DAE system, as we show in this paper, is the continuous-time counterpart of the link transmission model (Yperman et al., 2005). In addition, we present a solution existence theory for the continuous-time network model and investigate continuous dependence of the solution on the initial data, a property known as well-posedness. We test the DAE system extensively on several small and large networks and demonstrate its numerical efficiency.
Mathematical Models and Methods in Applied Sciences, 2017
We consider nonlinear transport equations with non-local velocity describing the time-evolution o... more We consider nonlinear transport equations with non-local velocity describing the time-evolution of a measure. Such equations often appear when considering the mean-field limit of finite-dimensional systems modeling collective dynamics. We address the problem of controlling these equations by means of a time-varying bounded control action localized on a time-varying control subset of small Lebesgue measure. We first define dissipativity for nonlinear transport equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, assuming that the uncontrolled system is dissipative, we provide an explicit construction of a control law steering the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function. In this sense the construction can be seen as an infinite-dimensional analogue of the well-known Jurdjevic–Quinn procedure. Moreover, the control l...
Transportation Research Part B: Methodological, 2016
This paper establishes the continuity of the path delay operators for dynamic network loading (DN... more This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill-Whitham-Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations (PDAEs). We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.
Automatica, 2017
For control-affine systems with a proper Lyapunov function, the classical procedure Jurdjevic-Qui... more For control-affine systems with a proper Lyapunov function, the classical procedure Jurdjevic-Quinn (see [21]) gives a well-known and widely used way of designing feedback controls that asymptotically stabilize the system to some invariant set. In this procedure, all controls are in general required to be activated at the same time. In this paper we give sufficient conditions under which this stabilization can be done by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. We thus obtain a sparse version of the classical Jurdjevic-Quinn theorem. We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying feedback, a sampled feedback, and a hybrid hysteresis. We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models.
Active Particles, Volume 1, 2017
In the present chapter we study the emergence of global patterns in large groups in first and sec... more In the present chapter we study the emergence of global patterns in large groups in first and second-order multi-agent systems, focusing on two ingredients that influence the dynamics: the interaction network and the state space. The state space determines the types of equilibrium that can be reached by the system. Meanwhile, convergence to specific equilibria depends on the connectivity of the interaction network and on the interaction potential. When the system does not satisfy the necessary conditions for convergence to the desired equilibrium, control can be exerted, both on finite-dimensional systems and on their mean-field limit.
Procedia - Social and Behavioral Sciences, 2012
Mathematical Control & Related Fields, 2013
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. B... more This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012. We focus on a controlled Cucker-Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues. We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which "sparse is better". We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
Mathematical Models and Methods in Applied Sciences, 2015
Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker–Smale (2007), t... more Starting with the seminal papers of Reynolds (1987), Vicsek et al. (1995), Cucker–Smale (2007), there has been a lot of recent works on models of self-alignment and consensus dynamics. Self-organization has so far been the main driving concept of this research direction. However, the evidence that in practice self-organization does not necessarily occur (for instance, the achievement of unanimous consensus in government decisions) leads to the natural question of whether it is possible to externally influence the dynamics in order to promote the formation of certain desired patterns. Once this fundamental question is posed, one is also faced with the issue of defining the best way of obtaining the result, seeking for the most "economical" way to achieve a certain outcome. Our paper precisely addressed the issue of finding the sparsest control strategy in order to lead us optimally towards a given outcome, in this case the achievement of a state where the group will be able...
Networks & Heterogeneous Media, 2006
We consider a mathematical model for fluid-dynamic flows on networks which is based on conservati... more We consider a mathematical model for fluid-dynamic flows on networks which is based on conservation laws. Road networks are considered as graphs composed by arcs that meet at some junctions. The crucial point is represented by junctions, where interactions occurr and the problem is underdetermined. The approximation of scalar conservation laws along arcs is carried out by using conservative methods, such as the classical Godunov scheme and the more recent discrete velocities kinetic schemes with the use of suitable boundary conditions at junctions. Riemann problems are solved by means of a simulation algorithm which proceeds processing each junction. We present the algorithm and its application to some simple test cases and to portions of urban network.
EMS Surveys in Mathematical Sciences, 2014
The broad research thematic of flows on networks was addressed in recent years by many researcher... more 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. 1 The author acknowledges partial support of 2013 GNAMPA project "Leggi di Conservazione: Teoria e Applicazioni". 2 The author acknowledges support by BMBF KinOpt, DFG Cluster of Excellence EXC128 and DAAD 54365630, 55866082. 3 The author acknowledges partial support of NSF Research Network in the Mathematical Sciences KI-Net "Kinetic description of emerging challenges in multiscale problems of natural sciences" Grant # : 1107444.
Contemporary Mathematics, 1999
In this paper we establish the existence of nonclassical entropy solutions for the Cauchy problem... more In this paper we establish the existence of nonclassical entropy solutions for the Cauchy problem associated with a conservation law having a nonconvex flux-function. Instead of the classical Oleinik entropy criterion, we use a single entropy inequality supplemented with a kinetic relation. We prove that these two conditions characterize a unique nonclassical Riemann solver. Then we apply the wave-front tracking method to the Cauchy problem. By introducing a new total variation functional, we can prove that the corresponding approximate solutions converge strongly to a nonclassical entropy solution.
SIAM Journal on Scientific Computing, 2011
Numerical techniques for the simulation of an ODE-PDE model for supply chains are presented. Firs... more Numerical techniques for the simulation of an ODE-PDE model for supply chains are presented. First we describe a scheme, based on Upwind and explicit Euler methods, provide corrections to maintain positivity of solutions, prove convergence and provide convergence rate. The latter is achieved via comparison with Wave Front Tracking solutions and the use of generalized tangent vectors. Different choice of time and space meshes give similar results, both for CPU times and numerical errors. Fast algorithms, based on an accurate choice of time and space meshes and data structures, are then proposed, achieving high computational gains.
SIAM Journal on Applied Mathematics, 2011
An extension of the Colombo phase transition model is proposed. The congestion phase is described... more An extension of the Colombo phase transition model is proposed. The congestion phase is described by a two-dimensional zone defined around an equilibrium flux known as the classical fundamental diagram. General criteria to build such a set-valued fundamental diagram are enumerated, and instantiated on several equilibrium fluxes with different concavity properties. The solution of the Riemann problem in the presence of phase transitions is obtained through the construction of a Riemann solver, which enables the definition of the solution of the Cauchy problem using wavefront tracking. The free-flow phase is described using a Newell-Daganzo fundamental diagram, which allows for a more tractable definition of phase transition compared to the original Colombo phase transition model. The accuracy of the numerical solution obtained by a modified Godunov scheme is assessed on benchmark scenarios for the different flux functions constructed.
SIAM Journal on Applied Dynamical Systems, 2008
In this paper we introduce a computation algorithm to trace car paths on road networks, whose loa... more In this paper we introduce a computation algorithm to trace car paths on road networks, whose load evolution is modeled by conservation laws. This algorithm is composed by two parts: computation of solutions to conservation equations on each road and localization of car position resulting by interactions with waves produced on roads. Some applications and examples to describe the behavior of a driver traveling in a road network are showed. Moreover a convergence result for wave front tracking approximate solutions, with BV initial data on a single road is established.