A computational method, based on l1-norm minimization, is proposed for the problem traffic flow correction. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corruptions accurately if there are only a few bad sensors which are located at certain links by the proposed method. We mathematically quantify these links that are robust to miscounts and relate them to the geometric structure of the traffic network. In a more realistic setting, besides the unhealthy link sensors, if small measure noise are present at the other sensors, our method guarantees to give an estimated traffic flow fairly close to the ground-truth. Both toy and real-world examples are provided to demonstrate the effectiveness of the proposed method.

We investigate, by numerical simulation, the shear layer instability associated with the outer layer of a spiral vortex formed behind an impulsively started thin ellipse. The unstable free shear layer undergoes a secondary instability. We connect this instability with the dynamics of corner vortices adjacent to the tip of the ellipse by observing that the typical turnover time of the corner vortex matches the period of the unstable mode in the shear layer. We suggest that the corner vortex acts as a signal generator, and produces periodic perturbation which triggers the instability.

The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa–Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa–Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy and momentum of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which support the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa–Holm equation under a wide class of initial data.

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound u (t,) u (t,)= O (1)(1+ t) | ln | on the distance between an exact BV solution u and a viscous approximation u, letting the viscosity coefficient 0. In the proof, starting from u we construct an approximation of the viscous solution u by taking a mollification u* and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed . Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. 2004 Wiley Periodicals, Inc.

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.