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We construct a model of traffic flow with sources and destinations on a roads network. The model is based on a conservation law for the density of traffic and on semilinear equations for traffic-type functions, i.e. functions describing paths for cars.
We propose a definition of solution at junctions, which depends on the traffic-type functions. Finally we prove, for every positive time T, existence of entropic solutions on the whole network for perturbations of constant initial data.
Our method is based on the wave-front tracking approach.
When numerically solving the Liouville equation with a discontinuous potential, one faces the problem of selecting a unique, physically relevant solution across the potential barrier, and the problem of a severe time step constraint due to the CFL condition. In this paper, we introduce two classes of Hamiltonian-preserving schemes for such problems. By using the constant Hamiltonian across the potential barrier, we introduce a selection criterion for a unique, physically relevant solution to the underlying linear hyperbolic equation with singular coefficients. These schemes have a hyperbolic CFL condition, which is a signicant improvement over a conventional discretization. These schemes are proved to be positive, and stable in both l∞ and l1 norms. Numerical experiments are conducted to study the numerical accuracy.
This work is motivated by the well-balanced kinetic schemes by Perthame and Simeoni for the shallow water equations with a discontinuous bottom topography, and has applications to the level set methods for the computations of multivalued physical observables in the semiclassical limit of the linear Schr%#x000F6;dinger equation with a discontinuous potential, among other applications.
We give an L2-well posedness result concerning an initial boundary value problem for the system of linear elasticity either in the half-plane or in a two dimensional bounded domain. Under the necessary uniform Kreiss Lopatinskii condition we construct here a dissipative Kreiss symmetrizer of our problem; actually, due to the characteristic boundary and the lack of a technical assumption given by T. Ohkubo, the main difficulty consists of building the dissipative symmetrizer near some special "boundary points".
Weeks method is a well established algorithm for the numerical inversion of scalar Laplace space functions. In this paper, we extend the method to the inversion of matrix functions of a single time variable and assess the qualities of this approach. To illustrate and quantify our discussion, we compute the matrix exponential by means of an FFT based algorithm. Particular attention is paid to a comparison of algorithms for the automated selection of two tuning parameters. In addition to selection algorithms from the literature, we introduce a pseudospectra based approach for the particular case of the matrix exponential. Finally, applications involving both pathological matrices and the numerical solution of differential equations highlight the utility of the method.
Recovering a function out of a finite number of moments is generally an ill-posed inverse problem. We focus on two special cases arising from applications to multiphase geometric optics computations where this problem can be carried out in a restricted class of given densities. More precisely, we present a simple algorithm for the inversion of Markov's moment problem which appears in the treatment of Brenier and Corrias' "K-multibranch solutions" and study Stieltje's algorithm in order to process moment systems arising from a Wigner analysis. Numerical results are provided for moderately intricate wave-fields.
A wide family of finite-difference methods for the linear advection equation, based on a six-point stencil, is presented. The family depends on three parameters and includes most of the classical linear schemes. A stability and consistency analysis is carried out. Numerical examples show the performance of the different methods according to the choice of the parameters. The problem of the determination of the parameters providing the "best approximation" is also addressed.
Constructing the visible and invisible regions of an observer due to the presence of obstacles in the environment has played a central role in many applications. It can also be a first step. In this paper, we adopt a visibility algorithm that can produce a variety of general information to handle the optimization of visibility information. Through the use of level set tools, gradient flow, finite differencing, and solvers for ordinary differential equations, we introduce a set of distinct algorithms for several model problems involving the optimization of visibility information.
Couplings of microscopic stochastic models to deterministic macroscopic ordinary and partial differential equations are commonplace in numerous applications such as catalysis, deposition processes, polymeric flows, biological networks and parametrizations of tropical and open ocean convection. In this paper we continue our study of the class of prototype hybrid systems presented in . These model systems are comprised of a microscopic Arrhenius dynamics stochastic process modeling adsorption/desorption of interacting particles which is coupled to an ordinary differential equation exhibiting a variety of bifurcation profiles. Here we focus on the case where phase transitions do not occur in the microscopic stochastic system and examine the influence of noise in the overall system dynamics.
Deterministic mean field and stochastic averaging closures derived in  are valid under stringent conditions on the range of microscopic interactions and time-scale separation respectively. Furthermore, their derivation is valid only for finite time intervals where rare events will not trigger a large deviation from the average behavior in the zero noise limit. In this paper we study such questions in the context of simple hybrid systems, demonstrating that deterministic closures based on various separation of scales arguments cannot in general capture transient and long-time dynamics. For this purpose we develop coarse grained stochastic closures for this class of hybrid systems and compare them to deterministic, mean-field and stochastic averaging closures. We show that the proposed coarse grained closures describe correctly the microscopic hybrid system solutions in all test cases examined, including rare events and random transitions between multiple stable states.