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We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black-Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.Mathematics Subject Classification (1991): 65M12, 35K55, 49L25

Abstract. In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this formulation the junctions disappear since each path is considered as a single uninterrupted road.We consider a Godunov-based approximation scheme for the system which is very easy to implement. Besides basic properties like the conservation of cars and positive bounded solutions, the scheme exhibits other nice properties, being able to select automatically a solution at junctions without requiring external procedures (e.g., maximization of the flux via a linear programming method). Moreover, the scheme can be interpreted as a discretization of the traffic models with buffer, although no buffer is introduced here.Finally, we show how the scheme can be recast in the framework of the classical theory of traffic flow on networks, where a conservation law has to be solved on each arc of the network. This is achieved by solving the Riemann problem for a modified equation, and showing that its solution corresponds to the one computed by the numerical scheme.

Abstract. In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence ad hoc external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.

We investigate finite difference schemes which approximate 2 × 2 one dimensional linear dissipative hyperbolic systems. We show that it is possible to introduce some suitable modifications in standard upwinding schemes, which keep into account the long-time behaviour of the solutions, to yield numerical approximations which are increasingly accurate for large times when computing small perturbations of stable asymptotic states, respectively around stationary solutions and in the diffusion (Chapman-Enskog) limit.1991 Mathematics Subject Classification. Primary: 65M12; Secondary: 35L65. Key words and phrases. Finite differences methods, dissipative hyperbolic problems, asymptotic behavior, asymptotic high order schemes, Chapman-Enskog expansion.

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 have developed a rat brain organotypic culture model, in which tissue slices contain cortex-subventricular zone-striatum regions, to model neuroblast activity in response to in vitro ischemia. Neuroblast activation has been described in terms of two main parameters, proliferation and migration from the subventricular zone into the injured cortex. We observed distinct phases of neuroblast activation as is known to occur after in vivo ischemia. Thus, immediately after oxygen/glucose deprivation (6–24 hours), neuroblasts reduce their proliferative and migratory activity, whereas, at longer time points after the insult (2 to 5 days), they start to proliferate and migrate into the damaged cortex. Antagonism of ionotropic receptors for extracellular ATP during and after the insult unmasks an early activation of neuroblasts in the subventricular zone, which responded with a rapid and intense migration of neuroblasts into the damaged cortex (within 24 hours). The process is further enhanced by elevating the production of the chemoattractant SDf-1α and may also be boosted by blocking the activation of microglia. This organotypic model which we have developed is an excellent in vitro system to study neurogenesis after ischemia and other neurodegenerative diseases. Its application has revealed a SOS response to oxygen/glucose deprivation, which is inhibited by unfavorable conditions due to the ischemic environment. Finally, experimental quantifications have allowed us to elaborate a mathematical model to describe neuroblast activation and to develop a computer simulation which should have promising applications for the screening of drug candidates for novel therapies of ischemia-related pathologies.

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.