Department of Mathematical Sciences and CCIB

Abstract New computation algorithms for a fluid-dynamic mathematical model of flows on networks are proposed, described and tested. First we improve the classical Godunov scheme (G) for a special flux function, thus obtaining a more efficient method, the Fast Godunov scheme (FG) which reduces the number of evaluations for the numerical flux. Then a new method, namely the Fast Shock Fitting method (FSF), based on good theorical properties of the solution of the problem is introduced.

Abstract We consider an hyperbolic conservation law with discontinuous flux. Such partial differential equation arises in different applicative problems, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions. Key-words: Conservation laws–discontinuous flux–Riemann Solvers–front tracking–traffic flow

The Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm.

We present a continuous-time link-based kinematic wave model (LKWM) for dynamic traffic networks based on the scalar conservation law model. 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. 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.

ABSTRACT We propose a bound-preserving Runge-Kutta (RK) discontinuous Galerkin (DG) method as an efficient, effective and compact numerical approach for numerical simulation of traffic flow problems on networks, with arbitrary high order accuracy. Road networks are modeled by graphs, composed of a finite number of roads that meet at junctions. On each road, a scalar conservation law describes the dynamics, while coupling conditions are specified at junctions to define flow separation or convergence at the points where roads meet. We incorporate such coupling conditions in the RK DG framework, and apply an arbitrary high order bound preserving limiter to the RK DG method to preserve the physical bounds on the network solutions (car density). We showcase the proposed algorithm on several benchmark test cases from the literature, as well as several new challenging examples with rich solution structures.

The broad research thematic of
ows on networks was addressed in recent years by
many researchers, in the area of applied mathematics, with new models based on partial
dierential equations. The latter brought a signicant innovation in a eld previously
dominated by more classical techniques from discrete mathematics or methods based on
ordinary dierential equations. In particular, a number of results, mainly dealing with
vehicular trac, supply chains and data networks, were collected in two monographs:
Trac
ow on networks, AIMSciences, Springeld, 2006, and Modeling, simulation,
and optimization of supply chains, SIAM, Philadelphia, 2010. The eld continues to

ourish 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
ow to
air trac management. The aim of the present survey paper is to present a view on large
number of themes, results and applications related to this broad research direction. The
authors cover dierent expertise (modeling, analysis, numeric, optimization and other)
so to provide an overview as extensive as possible. The focus is mainly on developments
appeared subsequently to the publication of the aforementioned books.

New computation algorithms for a
uid-dynamic mathematical model of

ows on networks are proposed, described and tested. First we improve
the classical Godunov scheme (G) for a special
ux function, thus obtaining
a more ecient method, the Fast Godunov scheme (FG) which reduces
the number of evaluations for the numerical
ux. Then a new method,
namely the Fast Shock Fitting method (FSF), based on good theorical
properties of the solution of the problem is introduced. Numerical results
and ecience tests are presented in order to show the behaviour of FSF
in comparison with G, FG and a conservative scheme of second order.