Electronic and Molecular Dynamics by the Quantum Wave Packet Method

A solution to the time‐dependent Schrödinger equation is required in a variety of problems in physics and chemistry. In this chapter, recent developments of numerical and theoretical techniques for quantum wave packet methods efficiently describe the dynamics of molecular dynamics, and electronic dynamics induced by ultrashort laser pulses in atoms and molecules will be reviewed, particularly on the development of grid methods and time‐propagation or pseudo‐time evolution methods developed recently. Applications of the quantum wave packet for studying the reactive resonances in F + H2/HD and O + O2 reaction, dissociative chemisorption of water on transition‐metal surfaces, state‐to‐state reaction dynamics, state‐to‐state tetra‐atomic reaction dynamics using transition wave packet method and reactant coordinate method, and electronic dynamics in H2+ and H2 molecules will be presented

A Weighted Residual Framework for Formulation and Analysis of Direct Transcription Methods for Optimal Control

In the past three decades, numerous methods have been proposed to transcribe optimal control problems (OCP) into nonlinear programming problems (NLP). In this dissertation work, a unifying weighted residual framework is developed under which most of the existing transcription methods can be derived by judiciously choosing test and trial functions. This greatly simplifies the derivation of optimality conditions and costate estimation results for direct transcription methods. Under the same framework, three new transcription methods are devised which are particularly suitable for implementation in an adaptive refinement setting. The method of Hilbert space projection, the least square method for optimal control and generalized moment method for optimal control are developed and their optimality conditions are derived. It is shown that under a set of equivalence conditions, costates can be estimated from the Lagrange multipliers of the associated NLP for all three methods. Numerical implementation of these methods is described using B-Splines and global interpolating polynomials as approximating functions. It is shown that the existing pseudospectral methods for optimal control can be formulated and analyzed under the proposed weighted residual framework. Performance of Legendre, Gauss and Radau pseudospectral methods is compared with the methods proposed in this research. Based on the variational analysis of first-order optimality conditions for the optimal control problem, an posteriori error estimation procedure is developed. Using these error estimates, an h-adaptive scheme is outlined for the implementation of least square method in an adaptive manner. A time-scaling technique is described to handle problems with discontinuous control or multiple phases. Several real-life examples were solved to show the efficacy of the h-adaptive and time-scaling algorithm

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Real-Space Mesh Techniques in Density Functional Theory

This review discusses progress in efficient solvers which have as their foundation a representation in real space, either through finite-difference or finite-element formulations. The relationship of real-space approaches to linear-scaling electrostatics and electronic structure methods is first discussed. Then the basic aspects of real-space representations are presented. Multigrid techniques for solving the discretized problems are covered; these numerical schemes allow for highly efficient solution of the grid-based equations. Applications to problems in electrostatics are discussed, in particular numerical solutions of Poisson and Poisson-Boltzmann equations. Next, methods for solving self-consistent eigenvalue problems in real space are presented; these techniques have been extensively applied to solutions of the Hartree-Fock and Kohn-Sham equations of electronic structure, and to eigenvalue problems arising in semiconductor and polymer physics. Finally, real-space methods have found recent application in computations of optical response and excited states in time-dependent density functional theory, and these computational developments are summarized. Multiscale solvers are competitive with the most efficient available plane-wave techniques in terms of the number of self-consistency steps required to reach the ground state, and they require less work in each self-consistency update on a uniform grid. Besides excellent efficiencies, the decided advantages of the real-space multiscale approach are 1) the near-locality of each function update, 2) the ability to handle global eigenfunction constraints and potential updates on coarse levels, and 3) the ability to incorporate adaptive local mesh refinements without loss of optimal multigrid efficiencies.Comment: 70 pages, 11 figures. To be published in Reviews of Modern Physic

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized Riemann Problems In Dispersive Hydrodynamics

Nonlinear, dispersive wave phenomena occur in a variety of physical contexts, both in nature and the laboratory. Mathematically, their dynamics can be modeled by a dispersive hydrodynamic system---a first order system of conservation laws modified by dispersion. In appropriate physical regimes, a multi-scale asymptotic expansion can by employed to derive a scalar equation from which we can infer approximate dynamics of the overarching system. We first study scalar models of dispersive hydrodynamics when dispersion is of higher order. Higher order dispersion in nonlinear, real-valued, local scalar equations can manifest when spatial derivatives are higher than third order. The primary mathematical framework we utilize is Whitham modulation theory, an asymptotic method to describe the slow modulations of a periodic wave's parameters. We identify three new classes of DSWs solutions to the Kawahara equation---a weakly nonlinear model that contains both third and fifth order dispersive terms. Numerical and asymptotic studies of the DSW solutions to the Kawahara equation motivate a further comprehensive study of the Whitham modulation equations for the fifth order Korteweg-de Vries (KdV5) equation. We compute various heteroclinic traveling waves that are shown to correspond to weak, discontinuous shock solutions of the KdV5-Whitham modulation system in the zero dispersion limit. The discontinuous shock solutions are shown to arise from a so-called generalized Riemann problem. The existence of heteroclinic traveling waves of the governing equation allow us to define the admissibility of discontinuous, weak solutions of the Whitham modulation equations, which we term Whitham shocks. Furthermore, the structure of the modulation equations, e.g. hyperbolicity or ellipticity, determine the modulational (in)stability of the heteroclinic traveling wave corresponding to an admissible Whitham shock. We then revisit the DSW solution of the KdV5 equation and demonstrate that it can be described in terms of a shock-rarefaction solution of the KdV5-Whitham modulation system. We conclude this portion with a discussion of how our results can be applied to other model dispersive hydrodynamic systems. We then investigate the interaction of a soliton and an evolving mean flow in bi-directional dispersive hydrodynamic media. The model equation is the defocusing nonlinear Schr&ouml;dinger equation. Utilizing Whitham modulation theory and posing a generalized Riemann problem for an initial jump in the mean flow and the soliton amplitude, a simple wave solution of the diagonalized NLS-Whitham modulation equations is obtained. This yields algebraic relationships between far-field initial data and the solitary wave amplitude from which we may infer the long-time soliton dynamics including the hydrodynamic trapping or transmission of the soliton by a, possibly, oscillatory mean flow.</p

plasma turbulence in magnetic fields

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