Photo-electron momentum spectra from minimal volumes: the time-dependent surface flux method

The time-dependent surface flux (t-SURFF) method is introduced for computing of strong-field infrared photo-ionization spectra of atoms by numerically solving the time-dependent Schr\"odinger equation on minimal simulation volumes. The volumes only need to accommodate the electron quiver motion and the relevant range of the atomic binding potential. Spectra are computed from the electron flux through a surface, beyond which the outgoing flux is absorbed by infinite range exterior complex scaling (irECS). Highly accurate infrared photo-electron spectra are calculated in single active electron approximation and compared to literature results. Detailed numerical evidence for performance and accuracy is given. Extensions to multi-electron systems and double ionization are discussed.Comment: 18 pages, 5 figure

Modeling and computation of Bose-Einstein condensates: stationary states, nucleation, dynamics, stochasticity

International audienceThe aim of this chapter is first to give an introduction to the derivation of the Gross-Pitaevskii Equations (GPEs) that arise in the modeling of Bose-Einstein Condensates (BECs). In particular, we describe some physical problems related to stationary states, dynamics, multi-components BECs and the possibility of handling stochastic effects into the equation. Next, we explain how to compute the stationary (and ground) states of the GPEs through the imaginary time method (also called Conjugate Normalized Gradient Flow) and finite difference or pseudo-spectral dis-cretization techniques. Examples are provided by using GPELab which is a Mat-lab toolbox dedicated to the numerical solution of GPEs. Finally, we explain how to discretize correctly the time-dependent GPE so that the schemes are physically admissible. We again provide some examples by using GPELab. Furthermore, extensions of the discretization schemes to some classes of stochastic (in time) GPEs are described and analyzed

Cumulative reports and publications through December 31, 1990

This document contains a complete list of ICASE reports. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Cumulative reports and publications through December 31, 1989

A complete list of reports from the Institute for Computer Applications in Science and Engineering (ICASE) is presented. The major categories of the current ICASE research program are: numerical methods, with particular emphasis on the development and analysis of basic numerical algorithms; control and parameter identification problems, with emphasis on effectual numerical methods; computational problems in engineering and the physical sciences, particularly fluid dynamics, acoustics, structural analysis, and chemistry; computer systems and software, especially vector and parallel computers, microcomputers, and data management. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available

Cumulative reports and publications

A complete list of Institute for Computer Applications in Science and Engineering (ICASE) reports are listed. Since ICASE reports are intended to be preprints of articles that will appear in journals or conference proceedings, the published reference is included when it is available. The major categories of the current ICASE research program are: applied and numerical mathematics, including numerical analysis and algorithm development; theoretical and computational research in fluid mechanics in selected areas of interest to LaRC, including acoustics and combustion; experimental research in transition and turbulence and aerodynamics involving LaRC facilities and scientists; and computer science

Trapped modes in non-uniform elastic waveguides: asymptotic and numerical methods

Trapped modes within elastic waveguides are investigated employing asymptotic and numerical methods. The problems considered in this thesis concentrate on linear elastic waves in thickened/thinned and curved waveguides. The localised modes are propagating within some region that is characterized by a small parameter but are cut-off for geometric reasons exterior to that region, and thereafter exponentially decay with distance along the waveguide. Given this physical interpretation long wave theories become appropriate. The general approach is as follows: an asymptotic scheme is developed to analyse whether trapped modes should be expected and to obtain the frequencies at which trapped modes are excited. The asymptotic approach leads to an ordinary differential equation eigenvalue problem that encapsulates the essential physics. Then, numerical simulations based on spectral methods are performed for this reduced equation and for the full elasticity equations to validate the asymptotic scheme and demonstrate its accuracy. The thesis begins with an investigation of trapping due to thickness variations. The long-wave model for trapped modes is derived and it is shown that this model is functionally the same as that for a bent plate. Careful computations of the exact governing equations are compared with the asymptotic theory to demonstrate that the theories tie together. Different boundary conditions upon the guide walls and the importance of the sign of the group velocity are discussed in detail. Then, it is shown that boundary conditions also play a crucial role in the possible existence of trapped modes. The possibility of trapped modes is considered in nonuniform elastic/ ocean/ quantum waveguides where the guide has one wall with Dirichlet (clamped) boundary conditions and the other Neumann (stress-free) boundary conditions. For bent waveguides, with such boundary conditions, the sign of the curvature function is shown to play an important role in the possibility of trapping. Trapped modes in 3D elastic plates are considered as a model of waves that are guided along, and localised to the vicinity of, welds. These waves propagate unattenuated along the weld and exponentially decay with distance transverse to it. Three-dimensional geometries introduce additional complications but, again, asymptotic analysis is possible. The long-wave model provides numerical values of the trapped mode frequencies and gives conditions at which trapping can occur; these depend on the components of the wave number in different directions and variations of the plate thickness. To mimic the guide stretching out to infinity a perfectly matched layer (PML) technique originally developed by Berenger for electromagnetic wave propagation is employed. The method is illustrated on the example of topographically varying and bent acoustic guides, and numerically implemented in the spectral scheme to construct dispersion curves for a two-dimensional circular elastic annulus immersed in infinite fluid. This numerical scheme is new and more efficient than direct root-finding methods for the exact dispersion relation involving the Bessel functions. In the final chapter, the influence of external fluid on trapping within elastic waveguides is considered. A long-wave scheme for a curved and thickening plates in infinite fluid is derived, conditions of existence of trapping are analysed and compared with those for plates in vacuum

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators

Computational Electromagnetism and Acoustics

It is a moot point to stress the significance of accurate and fast numerical methods for the simulation of electromagnetic fields and sound propagation for modern technology. This has triggered a surge of research in mathematical modeling and numerical analysis aimed to devise and improve methods for computational electromagnetism and acoustics. Numerical techniques for solving the initial boundary value problems underlying both computational electromagnetics and acoustics comprise a wide array of different approaches ranging from integral equation methods to finite differences. Their development faces a few typical challenges: highly oscillatory solutions, control of numerical dispersion, infinite computational domains, ill-conditioned discrete operators, lack of strong ellipticity, hysteresis phenomena, to name only a few. Profound mathematical analysis is indispensable for tackling these issues. Many outstanding contributions at this Oberwolfach conference on Computational Electromagnetism and Acoustics strikingly confirmed the immense recent progress made in the field. To name only a few highlights: there have been breakthroughs in the application and understanding of phase modulation and extraction approaches for the discretization of boundary integral equations at high frequencies. Much has been achieved in the development and analysis of discontinuous Galerkin methods. New insight have been gained into the construction and relationships of absorbing boundary conditions also for periodic media. Considerable progress has been made in the design of stable and space-time adaptive discretization techniques for wave propagation. New ideas have emerged for the fast and robust iterative solution for discrete quasi-static electromagnetic boundary value problems

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

Numerical scalar curvature deformation and a gluing construction

In this work a new numerical technique to prepare Cauchy data for the initial value problem (IVP) formulation of Einstein's field equations (EFE) is presented. Our method is directly inspired by the exterior asymptotic gluing (EAG) result of Corvino (2000). The argument assumes a moment in time symmetry and allows for a composite, initial data set to be assembled from (a finite subdomain of) a known asymptotically Euclidean initial data set which is glued (in a controlled manner) over a compact spatial region to an exterior Schwarzschildean representative. We demonstrate how (Corvino, 2000) may be directly adapted to a numerical scheme and under the assumption of axisymmetry construct composite Hamiltonian constraint satisfying initial data featuring internal binary black holes (BBH) glued to exterior Schwarzschild initial data in isotropic form. The generality of the method is shown in a comparison of properties of EAG composite initial data sets featuring internal BBHs as modelled by Brill-Lindquist and Misner data. The underlying geometric analysis character of gluing methods requires work within suitably weighted function spaces, which, together with a technical impediment preventing (Corvino, 2000) from being fully constructive, is the principal difficulty in devising a numerical technique. Thus the single previous attempt by Giulini and Holzegel (2005) (recently implemented by Doulis and Rinne (2016)) sought to avoid this by embedding the result within the well known Lichnerowicz-York conformal framework which required ad-hoc assumptions on solution form and a formal perturbative argument to show that EAG may proceed. In (Giulini and Holzegel, 2005) it was further claimed that judicious engineering of EAG can serve to reduce the presence of spurious gravitational radiation - unfortunately, in line with the general conclusion of (Doulis and Rinne, 2016) our numerical investigation does not appear to indicate that this is the case. Concretising the sought initial data to be specified with respect to a spatial manifold with underlying topology R×S² our method exploits a variety of pseudo-spectral (PS) techniques. A combination of the eth-formalism and spin-weighted spherical harmonics together with a novel complex-analytic based numerical approach is utilised. This is enabled by our Python 3 based numerical toolkit allowing for unified just-in-time compiled, distributed calculations with seamless extension to arbitrary precision for problems involving generic, geometric partial differential equations (PDE) as specified by tensorial expressions. Additional features include a layer of abstraction that allows for automatic reduction of indicial (i.e., tensorial) expressions together with grid remapping based on chart specification - hence straight-forward implementation of IVP formulations of the EFE such as ADM-York or ADM-York-NOR is possible. Code-base verification is performed by evolving the polarised Gowdy T³ space-time with the above formulations utilising high order, explicit time-integrators in the method of lines approach as combined with PS techniques. As the initial data we prepare has a precise (Schwarzschild) exterior this may be of interest to global evolution schemes that incorporate information from spatial-infinity. Furthermore, our approach may shed light on how more general gluing techniques could potentially be adapted for numerical work. The code-base we have developed may also be of interest in application to other problems involving geometric PDEs