High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton's method. Our numerical simulations show that Newton's method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities.The one-dimensional results of the current paper create a foundation for the analysis of multi-dimensional problems in the future.Comment: 47 pages, 8 figure

Estudio y construcción de métodos de integración numérica de tipo Runge-Kutta-Hermite-Birkhoff

En este trabajo fin de grado se pretende el análisis y construcción de un tipo particular de integradores numéricos más general que los métodos Runge-Kutta. El objetivo es el desarrollo de un código de integración numérica a paso variable de este tipo de esquemas con especial atención a las propiedades de dispersión y disipación numéricas

Numerical and Analytical Studies of Electromagnetic Waves: Hermite Methods, Supercontinuum Generation, and Multiple Poles in the SEM

The dissertation consists of three parts: Hermite methods, scattering from a lossless sphere, and analysis of supercontinuum generation. Hermite methods are a new class of arbitrary order algorithms to solve partial differential equations (PDE). In the first chapter, we discuss the fundamentals of Hermite methods in great detail. Hermite interpolation is discussed as well as the different time evolution schemes including Hermite-Taylor and Hermite-Runge-Kutta schemes. Further, an order adaptive Hermite method for initial value problems is described. Analytical studies and numerical simulations in both 1D and 2D are presented. To handle geometry, a hybrid Hermite discontinuous Galerkin method is introduced. A discontinuous Galerkin method is used next to the boundaries to handle the geometry and the boundary conditions, while a Hermite method is used in the interior of the computation domain to enhance the performance. Numerical simulations of 1D wave propagation and the solutions to 2D Maxwell\u27s TM equations are presented along with performance and accuracy data. In the second chapter, we study the scattering problem concerning the scattering poles from a lossless sphere for both acoustic and electromagnetic waves. We show that in certain cases there exist only first order scattering poles, but in some other cases, arbitrary order scattering poles can be found by imposing certain lossless impedance boundary conditions on the spherical scatterer. A method to construct arbitrary order scattering poles is discussed. The impedance loading function is required to satisfy Foster\u27s theorem so that the scattering problem is lossless. In the last chapter, we analyse the generation of supercontinua in photonic crystal fibers. We depart from the commonly used approach where a Taylor series expansion of the propagation constant is used to model the dispersive properties in a generalized nonlinear Schrodinger equation (gNLSE). Instead, we develop a mathematical model starting from numerically calculated group velocity dispersion (GVD) curves. Then, we construct a certain function over a broad frequency window and integrate the gNLSE in a way so that the spectral dependence of the propagation constant is preserved. We found that the generation of broadband supercontinua in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. Numerical simulations show that if the nonlinear self-steepening is strong enough, the model produces a shock that is not arrested by dispersion, whereas for weaker nonlinearity the pulse propagates the full extent of the fiber with the generation of a supercontinuum

Geometric Numerical Integration

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions