Theoretical and numerical studies of some problems in reaction-diffusion equations, electromagnetics and statistical modeling of turbulent flows.

Abstract

This thesis investigates three problems of applied mathematics. The problems are unrelated to each other. It is the underlying theory that gives them a common denominator. The first part of the thesis examines the behavior of a chaotic system when one adds diffusion to it. More specifically we examine a system of three reaction diffusion equations (with one spatial dimension) where the reaction term is the usual Lorenz system. We are interested in the dynamics of this system with periodic boundary conditions as the diffusion parameter goes to zero. We prove that the system admits an invariant region, and has a unique solution for all initial data in the invariant region. We study the stability of the trivial solution and discover a sequence of simple bifurcation points. We follow the new solutions numerically and sketch a diagram in parameter space for a fixed value of the diffusion parameter and different values of the Lorenz parameter ρ. We construct asymptotic expansions to understand the basic dynamics of the equations. We discuss the difficulties of creating a consistent asymptotic expansion. Finally we present an efficient and accurate way to simulate the evolution of the system numerically. The second part offers a novel way to solve numerically Maxwell's equations in a two dimensional parallel periodic waveguide. The method we propose is spectrally accurate in the direction of propagation and second order accurate in the other directions. It really is a variant of a well known and used method called the Finite Difference Time Domain (FDTD) method. We calculate the CFL condition for the method and do a phase error analysis for errors occurring due to the finite differencing in the non periodic and temporal directions. We conclude that the phase error is mainly due to the spatial discretization in the transverse direction. We discuss two different ways to extract the frequency ω from a numerical simulation for a given wavenumber β. Our results show excellent agreement with cases where the answer is known either analytically or experimentally. The last part of the thesis presents a new way to approach turbulent flows. The idea is to write an equation for the Probability Density Function (PDF) of a dynamical system with noise. If one assumes that the PDF depends on a finite number of parameters, we seek for ordinary differential equations for the time evolution of these parameters. We present the theory and its implementation on an one, three and five dimensional example. We also discuss its implementation for the Navier-Stokes equations in two dimensions for a periodic box and channel flow. We point out the advantages and disadvantages of various PDFs and discuss how one can use variations of a PDF for the needs of a particular problem.