Two combined methods for the global solution of implicit semilinear differential equations with the use of spectral projectors and Taylor expansions

Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations

Algorithms for Analysis of Nonlinear High-Frequency Circuits

The most efficient simulation solvers use composite procedures that adaptively rearrange computation algorithms to maximize simulation performance. Fast and stable processing optimized for given simulation problem is essential for any modern simulator. It is characteristic for electronic circuit analysis that complexity of simulation is affected by circuit size and used device models. Implementation of electronic device models in program SPICE uses traditional implementation allowing fast computation but further modification of model can be questionable. The first fundamental thesis aim is scalability of the simulation based on the adaptive internal solver composing different algorithms according to properties of simulation problem to maximize simulation performance. In a case of the small circuit as faster solution prove simple, straightforward methods that utilize arithmetic operations without unnecessary condition jumping and memory rearrangements that can not be effectively optimized by a compiler. The limit of small size simulation problems is related to computation machine capabilities. The present day PC sets this limit to fifty independent voltage nodes where inefficiency of calculation procedure does not play any role in overall processor performance. The scalable solver must also be able to handle correctly simulation of large-scale circuits that requires entirely different approach apart to standard size circuits. The unique properties of simulation of the electronic circuits that played until this time only the minor role suddenly gain on significance for circuits with several thousand voltage nodes. In those particular cases, iterative algorithms based on Krylov subspace methods provide better results from the aspect of performance than standard direct methods. This thesis also proposes unique techniques of indexation of the large-scale sparse matrix system. The primary purpose is to reduce memory requirements for storing sparse matrices during simulation computation. The second fundamental thesis aim is automatic adaptivity of device models definition respecting current simulation state and settings. This principle is denoted as Functional Chaining mechanism that is based on the principle of the automatic self-modifying procedure utilizing state-of-the-art functional computation layer during the simulation process. It can significantly improve mapping performance of circuit variables to device models; it also allows autonomous redefinition of simulation algorithms during analysis with an intention to reduce computation time. The core idea is based on utilization of programming principles related to functional programming languages. It is also presents possibilites of reimplementation to the modern object-oriented languages. The third fundamental thesis aim focuses on simulation accuracy and reliability. Arbitrary precision variable types can directly lead to increased simulation accuracy but on the other hand; they can significantly decrease simulation performance. In last chapters, there are several algorithms provided with the claim to provide better simulation accuracy and suppress computation errors of floating point data types.Katedra radioelektronik

TDDFT Studie des Energie- und Ladungstransfers bei der Wechselwirkung von Ionen mit Festkörperoberflächen

The investigation of processes at the interface between a plasma and a solid surface is of high interest to many different research areas with a vast variety of technical applications. These extend from electronics, surface chemistry and fusion research, to medicine, pharmacy and many more. However, an accurate understanding of the fundamental processes at the plasma-solid interface is often missing. One approach towards a more detailed understanding is to isolate the interaction of atoms, ions, or molecules with the surface and perform scattering experiments. Of special interest are low energy and hyperthermal projectiles with kinetic energies starting in the order of chemical binding energies (in the order of 1 eV) where projectiles are too slow to destroy or sputter the surface but contribute themselves enough energy to cause notable excitations within the electronic system. In this work, time-dependent density-functional theory (TDDFT) simulations have been employed together with Ehrenfest molecular dynamics (MD) as implemented in the Octopus code to investigate the charge and energy transfer between hyperthermal ions and metal surfaces. The focus of this work lies on the resonant neutralization of protons (H+) interacting with an Al(111) surface. The surface is modeled by a cluster geometry. Of major importance to TDDFT-MD simulations is the the quality of the approximation applied to the exchange-correlation potential which is investigated and discussed in detail. Also, difficulties connected with the spin-polarization within the calculations are analyzed. Furthermore, the neutralization process is studied, neutralization distances are determined and the energy transfer into electronic, kinetic and phononic degrees of freedom is analyzed. It has been found that the difference in initial kinetic energy between H+ and H0 projectiles ..

High harmonic generation in gas phase and condensed matter

The thesis presented here is comprised of two major investigations into strong field processes resulting from interaction between femtosecond laser pulses and atoms, either in isolation or assembled into a crystalline solid form. Our theoretical approach throughout these investigations is based on the numerical solution of the time-dependent Schrodinger equation (TDSE) using massively parallelised, high-performance computational tools. In particular, Chapter 2 explores high harmonic generation (HHG) in gas targets, a non-linear process in which an electron escapes an atomic system via tunnelling only to recombine with its parent system due to the oscillation of the interacting laser pulse and release a harmonic photon multiple of the driving laser frequency. In our gas HHG investigations we simulate high harmonic spectra using an efficient single-active-electron (SAE) TDSE treatment for noble gases and transitional metals, and introduce a novel efficient approach to modelling electron correlation processes in HHG, a multiplicative correlation enhancement factor (CEF) constructed as a ratio of photo-ionisation cross-sections (PICS) under the Hartree-Fock Approximation (HFA), without correlations, and the Random Phase Approximation with Exchange (RPAE), with correlations. We demonstrate the success of the adopted TDSE treatment for HHG with krypton and xenon targets where we reproduce the Cooper minimum at ~ 88 eV for the former and demonstrate the giant autoionizing resonance (GAR) at 100 eV using our CEF approach for the latter. We also consider xenon in a two-colour setup and demonstrate the success of the CEF approach in more complex laser pulse situations, reproducing the observed enhancement across varying relative phase. Our results across our HHG simulations for gas targets provide us with a solid foundation for further investigation into HHG in transition metals and solids. In Section 2.2.3 we demonstrate the application of the correlation enhancement approach to transition metal Mn and its ionic species Mn+, successfully reproducing the giant autoionizing resonance in both at 50 eV as reported in experiment and demonstrating the significant difference between ionic Mn+ and Mn in the contributions of the 4s and 3d_m=0 initial states for 400 nm calculations. Extending our investigation of HHG to solid targets, Chapter 3 explores a new simulation technique using time-dependent density functional theory (TDDFT) to model the richer dynamics of inter- and intra-band harmonic generation. We adopt an ab initio approach to model high harmonic generation, through the SALMON-TDDFT program and explore thin film semiconductors in line with recent work by collaborators. In particular, we review literature for diamond and establish clear harmonic structure where previously propagation and dephasing techniques had been required to resolve theory and experiment. We consider long 200 fs full duration pulses at 800 nm wavelengths and review recent literature regarding bulk silicon at 3000 nm and demonstrate clear harmonics at 2000 and 3000 nm, failing to observe any joint density of states (JDOS) effect and subsequent noisy-to-clean harmonic transition. We also demonstrate the effect of electromagnetic propagation through Maxwell+TDDFT calculations for thick samples of silicon, finding a pronounced effect as noted by Floss et al. previously. Finally we demonstrate the importance of the dephasing effect through a first-of-its-kind molecular dynamics simulations for silicon, without requiring phenomenological relaxation parameter T_2, and suggest helium or liquid nitrogen cooling of solid targets could improve harmonic returns. Finally we summarise and consider the future research based on the body of work presented here in Chapter 4

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 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