Convergence of numerical schemes for short wave long wave interaction equations

We consider the numerical approximation of a system of partial differential equations involving a nonlinear Schr\"odinger equation coupled with a hyperbolic conservation law. This system arises in models for the interaction of short and long waves. Using the compensated compactness method, we prove convergence of approximate solutions generated by semi-discrete finite volume type methods towards the unique entropy solution of the Cauchy problem. Some numerical examples are presented.Comment: 31 pages, 7 figure

A hyperbolic conservation law and particle systems

In these notes we consider two particle systems: the totally asymmetric simple exclusion process and the totally asymmetric zero-range process. We introduce the notion of hydrodynamic limit and describe the partial differential equation that governs the evolution of the conserved quantity – the density of particles p(t,.). This equation is a hyperbolic conservation law of type ətp(p, u) + vF(p(t, u)) = 0, where the flux F is a concave function. Taking these systems evolving on the Euler time scale tN, a central limit theorem for the empirical measure holds and the temporal evolution of the limit density field is deterministic. By taking the system in a reference frame with constant velocity, the limit density field does not evolve in time. In order to have a non-trivial limit, time needs to be speeded up and for time scales smaller than tN 4=3, there is still no temporal evolution. As a consequence, the current across a characteristic vanishes up to this longer time scale.Fundação para a Ciência e a Tecnologia (FCT) - bolsa SFRH/BPD/39991/2007Fundação Calouste Gulbenkian - projecto "Hydrodynamic limit of particle systems

Binary-coded and real-coded genetic algorithm in pipeline flow optimization

The mathematical model for the liquid-gas mixture flow in pipelines is an initial-boundary value problem for a nonlinear hyperbolic conservation law system. This hyperbolic conservation law system together with boundary conditions is numerically solved using the essentially non-oscillatory (ENO) schemes. The optimization problem is a boundary control problem, i.e. boundary conditions that cause pressure values in the pipeline as close as possible to the desired ones are to be found, considering given constraints. The applied optimization method is the genetic algorithm (GA) with two different variable-to-chromosome coding strategies: binary coding and real coding. The results of both GA strategies applied to two pipeline flow optimization problems are presented and compared