The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation-dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta impacts deeply on the quality of the solution. We analyze one-dimensional dissipation-dispersion to select the best combination of the space-time discretization for high Courant numbers. Then, we apply our findings to the integration of one-dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.

This paper addresses the numerical modeling of isentropic gas dynamics on one-dimensional networks, focusing on the Euler equations with a novel class of transmission conditions at network junctions, termed the Jump Transmission Condition (JTC). Unlike traditional models that enforce continuity of density at junctions, the (JTC) allows for a density jump proportional to the flux, reflecting phenomena such as bottlenecks in biological or traffic networks. The authors propose a relaxation scheme based on the vector BGK approach, which ensures mass conservation and enforces the (JTC) without requiring the explicit solution of the Riemann problem at the junction. The scheme is analyzed for its mathematical properties, including entropy dissipation and positivity, and is compared with classical solvers based on the explicit solution of the Riemann problem. Numerical experiments on simple and complex networks demonstrate the accuracy, robustness, and flexibility of the proposed method, especially in handling both subsonic and supersonic regimes and in extending to general networks.

Generalized Wasserstein distances allow us to quantitatively compare two continuous or atomic mass distributions with equal or different total masses. In this paper, we propose four numerical methods for the approximation of three different generalized Wasserstein distances introduced in the past few years, giving some insights into their physical meaning. After that, we explore their usage in the context of a sensitivity analysis of differential models for traffic flow. The quantification of the models’ sensitivity is obtained by computing the generalized Wasserstein distances between two (numerical) solutions corresponding to different inputs, including different boundary conditions.

In this paper, we present an extension of the Generic Second Order Models (GSOM) for traffic flow on road networks. We define a Riemann solver at the junction based on a priority rule and provide an iterative algorithm to construct solutions at junctions with n incoming and m outgoing roads. The logic underlying our solver is as follows: the flow is maximized while respecting the priority rule, which can be adjusted if the supply of an outgoing road exceeds the demand of a higher-priority incoming road. Approximate solutions for Cauchy problems are constructed using wave-front tracking. We establish bounds on the total variation of waves interacting with the junction and present explicit calculations for junctions with two incoming and two outgoing roads. A key novelty of this work is the detailed analysis of returning waves - waves generated at the junction that return to the junction after interacting along the roads - which, in contrast to first-order models such as LWR, can increase flux variation.

In this paper, we develop new methods to join machine learning techniques and macroscopic differential models, aimed at estimate and forecast vehicular traffic. This is done to complement respective advantages of data-driven and model-driven approaches. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: (1) if congestion is appearing under the sensor, and (2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident/bottleneck happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better predict the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A.

Current research directions indicate that vehicles with autonomous capabilities will increase in traffic contexts. Starting from data analyzed in R. E. Stern et al. (2018), this paper shows the benefits due to the traffic control exerted by a unique autonomous vehicle circulating on a ring track with more than 20 human-driven vehicles. Considering different traffic experiments with high stop-and-go waves and using a general microscopic model for emissions, it was first proved that emissions reduces by about 25%. Then, concentrations for pollutants at street level were found by solving numerically a system of differential equations with source terms derived from the emission model. The results outline that ozone and nitrogen oxides can decrease, depending on the analyzed experiment, by about 10% and 30%, respectively. Such findings suggest possible management strategies for traffic control, with emphasis on the environmental impact for vehicular flows.

In this paper we provide emission estimates due to vehicular traffic via Generic Second OrderModels. We generalize them to model road networks with merge and diverge junctions. Theprocedure consists on solving the Riemann Problem at junction assuming the maximization ofthe flow and a priority rule for the incoming roads. We provide some numerical results for asingle-lane roundabout and we propose an application of the given procedure to estimate theproduction of nitrogen oxides (NOx) emission rates. In particular, we show that the presence ofa traffic lights produces a 28% increase in the NOx emissions with respect to the roundabout.

In this paper we propose a modeling setting and a numerical Riemann problem solver at the junction of one dimensional shallow-water channel networks. The junction conditions take into account the angles with which the channels intersect and include the possibility of channels with different sections. The solver is illustrated with several numerical tests which underline the importance of the angle dependence to obtain reliable solutions.

The societal impact of traffic is a long-standing and complex problem. We focus on the estimation of ground-level ozone production due to vehicular traffic. We propose a comprehensive computational approach combining four consecutive modules: a traffic simulation module, an emission module, a module for the main chemical reactions leading to ozone production, and a module for the diffusion of gases in the atmosphere. The traffic module is based on a second-order traffic flow model, obtained by choosing a special velocity function for the Collapsed Generalized Aw-Rascle-Zhang model. A general emission module is taken from literature, and tuned on NGSIM data together with the traffic module. Last two modules are based on reaction-diffusion partial differential equations. The system of partial differential equations describing the main chemical reactions of nitrogen oxides presents a source term given by the general emission module applied to the output of the traffic module. We use the proposed approach to analyze the ozone impact of various traffic scenarios and describe the effect of traffic light timing. The numerical tests show the negative effect of vehicles restarts on emissions, and the consequent increase in pollutants in the air, suggesting to increase the length of the green phase of traffic lights.

In this paper we propose a multiscale traffic model, based on the family of Generic Second Order Models, which integrates multiple trajectory data into the velocity function. This combination of a second order macro- scopic model with microscopic information allows us to reproduce significant variations in speed and acceleration that strongly influence traffic emissions. We obtain accurate approximations even with a few trajectory data. The pro- posed approach is therefore a computationally efficient and highly accurate tool for calculating macroscopic traffic quantities and estimating emissions.

In this paper, we propose two models describing the dynamics of heavy and light vehicles on a road network, taking into account the interactions between the two classes. The models are tailored for two-lane highways where heavy vehicles cannot overtake. This means that heavy vehicles cannot saturate the whole road space, while light vehicles can. In these conditions, the creeping phenomenon can appear, i.e., one class of vehicles can proceed even if the other class has reached the maximal density. The first model we propose couples two first-order macroscopic LWR models, while the second model couples a second-order microscopic follow-the-leader model with a first-order macroscopic LWR model. Numerical results show that both models are able to catch some second-order (inertial) phenomena such as stop and go waves. Models are calibrated by means of real data measured by fixed sensors placed along the A4 Italian highway Trieste-Venice and its branches, provided by Autovie Venete S.p.A.

We study the rate of weak convergence of Markov chains to diffusion processes under quite general assumptions. We give an example in the financial framework, applying the convergence analysis to a multiple jumps tree approximation of the CIR process. Then, we combine the Markov chain approach with other numerical techniques in order to handle the different components in jump- diffusion coupled models. We study the analytical speed of convergence of this hybrid approach and provide an example in finance, applying our results to a tree-finite difference approximation in the Heston and Bates models. © 2021 Society for Industrial and Applied Mathematics.

The societal impact of traffic is a long-standing and complex problem. We focus on the estimation of ground-level ozone production due to vehicular traffic. We propose a comprehensive computational approach combining four consecutive modules: a traffic simulation module, an emission module, a module for the main chemical reactions leading to ozone production, and a module for the diffusion of gases in the atmosphere. The traffic module is based on a second-order traffic flow model, obtained by choosing a special velocity function for the Collapsed Generalized Aw-Rascle-Zhang model. A general emission module is taken from literature, and tuned on NGSIM data together with the traffic module. Last two modules are based on reaction-diffusion partial differential equations. The system of partial differential equations describing the main chemical reactions of nitrogen oxides presents a source term given by the general emission module applied to the output of the traffic module. We use the proposed approach to analyze the ozone impact of various traffic scenarios and describe the effect of traffic light timing. The numerical tests show the negative effect of vehicles restarts on emissions, and the consequent increase in pollutants in the air, suggesting to increase the length of the green phase of traffic lights.

The aim of this paper is to solve an inverse problem which regards a mass moving in a bounded domain. We assume that the mass moves following an unknown velocity field and that the evolution of the mass density can be described by a partial differential equation, which is also unknown. The input data of the problems are given by some snapshots of the mass distribution at certain times, while the sought output is the velocity field that drives the mass along its displacement. To this aim, we put in place an algorithm based on the combination of two methods: first, we use the dynamic mode decomposition to create a mathematical model describing the mass transfer; second, we use the notion of Wasserstein distance (also known as earth mover's distance) to reconstruct the underlying velocity field that is responsible for the displacement. Finally, we consider a real-life application: the algorithm is employed to study the travel flows of people in large populated areas using, as input data, density profiles (i.e., the spatial distribution) of people in given areas at different time instants. These kinds of data are provided by the Italian telecommunication company TIM and are derived by mobile phone usage.

The impact of vehicular traffic on society is huge and multifaceted, including economic, social, health and environmental aspects. The problems is complex and hard to model since it requires to consider traffic patterns, air pollutant emissions, and the chemical reactions and dynamics of pollutants in the low atmosphere. This paper aims at exploring a comprehensive simulation tool ranging from vehicular traffic all the way to environmental impact. As first step in this direction, we couple a traffic second-order model, tuned on NGSIM data, with an nitrogen oxides (NO) emission model and a set of equations for some of the main chemical reactions behind ozone (O) production.

We develop and study stability properties of a hybrid approximation of functionals of the Bates jump model with stochastic interest rate that uses a tree method in the direction of the volatility and the interest rate and a finite-difference approach in order to handle the underlying asset price process. We also propose hybrid simulations for the model, following a binomial tree in the direction of both the volatility and the interest rate, and a space-continuous approximation for the underlying asset price process coming from a Euler-Maruyama type scheme. We test our numerical schemes by computing European and American option prices.

In this paper we investigate the sensitivity of the LWR model on network to its parameters and to the network itself. The quantification of sensitivity is obtained by measuring the Wasserstein distance between two LWR solutions corresponding to different inputs. To this end, we propose a numerical method to approximate the Wasserstein distance between two density distributions defined on a network. We found a large sensitivity to the traffic distribution at junctions, the network size, and the network topology.

Network flows and specifically water flow in open canals can be modeled by systems of balance laws defined on graphs. The shallow water or Saint-Venant system of balance laws is one of the most used model and present two phases: fluvial or sub-critical and torrential or super-critical. Phase transitions may occur within the same canal but transitions related to networks are less investigated. In this paper we provide a complete characterization of possible phase transitions for a case study of a simple scenario with two canals and one junction. However, our analysis allows the study of more complicate networks. Moreover, we provide some numerical simulations to show the theory at work.

In this paper we deal with the study of travel flows and patterns of people in large populated areas. Information about the movements of people is extracted from coarse-grained aggregated cellular network data without tracking mobile devices individually. Mobile phone data are provided by the Italian telecommunication company TIM and consist of density profiles (i.e. the spatial distribution) of people in a given area at various instants of time. By computing a suitable approximation of the Wasserstein distance between two consecutive density profiles, we are able to extract the main directions followed by people, i.e. to understand how the mass of people distribute in space and time. The main applications of the proposed technique are the monitoring of daily flows of commuters, the organization of large events, and, more in general, the traffic management and control.

We tackle the issue of measuring and understanding the visitors' dynamics in a crowded museum in order to create and calibrate a predictive mathematical model. The model is then used as a tool to manage, control and optimize the fruition of the museum. Our contribution comes with one successful use case, the Galleria Borghese in Rome, Italy.