A family of high-order multistep methods with vanished phase-lag and its derivatives for the numerical solution of the Schrödinger equation

AbstractMany simulation algorithms (chemical reaction systems, differential systems arising from the modelling of transient behaviour in the process industries etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge–Kutta single-step methods are used. For the simulation of chemical procedures the radial Schrödinger equation is used frequently. In the present paper we will study a class of linear multistep methods. More specifically, the purpose of this paper is to develop an efficient algorithm for the approximate solution of the radial Schrödinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. Hence the main result of this paper is the development of an efficient multistep method for the numerical solution of systems of ordinary differential equations with oscillating or periodical solutions. The reason of their efficiency, as the analysis proved, is that the phase-lag and its derivatives are eliminated. Another reason of the efficiency of the new obtained methods is that they have high algebraic orde

Systematic construction of efficient six-stage fifth-order explicit Runge-Kutta embedded pairs without standard simplifying assumptions

This thesis examines methodologies and software to construct explicit Runge-Kutta (ERK) pairs for solving initial value problems (IVPs) by constructing efficient six-stage fifth-order ERK pairs without standard simplifying assumptions. The problem of whether efficient higher-order ERK pairs can be constructed algebraically without the standard simplifying assumptions dates back to at least the 1960s, with Cassity's complete solution of the six-stage fifth-order order conditions. Although RK methods based on the six-stage fifth-order order conditions have been widely studied and have continuing practical importance, prior to this thesis, the aforementioned complete solution to these order conditions has no published usage beyond the original series of publications by Cassity in the 1960s. The complete solution of six-stage fifth-order ERK order conditions published by Cassity in 1969 is not in a formulation that can easily be used for practical purposes, such as a software implementation. However, it is shown in this thesis that when the order conditions are solved and formulated appropriately using a computer algebra system (CAS), the generated code can be used for practical purposes and the complete solution is readily extended to ERK pairs. The condensed matrix form of the order conditions introduced by Cassity in 1969 is shown to be an ideal methodology, which probably has wider applicability, for solving order conditions using a CAS. The software package OCSage developed for this thesis, in order to solve the order conditions and study the properties of the resulting methods, is built on top of the Sage CAS. However, in order to effectively determine that the constructed ERK pairs without standard simplifying assumptions are in fact efficient by some well-defined criteria, the process of selecting the coefficients of ERK pairs is re-examined in conjunction with a sufficient amount of performance data. The pythODE software package developed for this thesis is used to generate a large amount of performance data from a large selection of candidate ERK pairs found using OCSage. In particular, it is shown that there is unlikely to be a well-defined methodology for selecting optimal pairs for general-purpose use, other than avoiding poor choices of certain properties and ensuring the error coefficients are as small as possible. However, for IVPs from celestial mechanics, there are obvious optimal pairs that have specific values of a small subset of the principal error coefficients (PECs). Statements seen in the literature that the best that can be done is treating all PECs equally do not necessarily apply to at least some broad classes of IVPs. By choosing ERK pairs based on specific values of individual PECs, not only are ERK pairs that are 20-30% more efficient than comparable published pairs found for test sets of IVPs from celestial mechanics, but the variation in performance between the best and worst ERK pairs that otherwise would seem to have similar properties is reduced from a factor of 2 down to as low as 15%. Based on observations of the small number of IVPs of other classes in common IVP test sets, there are other classes of IVPs that have different optimal values of the PECs. A more general contribution of this thesis is that it specifically demonstrates how specialized software tools and a larger amount of performance data than is typical can support novel empirical insights into numerical methods

Simulation of axisymmetric stepped surfaces with a facet

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 255-266).A crystal lattice with a small miscut from the plane of symmetry has a surface which consists of a series of atomic height steps separated by terraces. If the surface of this crystal is not in equilibrium with the surrounding medium, then its evolution is strongly mediated by the presence of these steps, which act as sites for attachment and detachment of diffusing adsorbed atoms ('adatoms'). In the absence of material deposition and evaporation, steps move in response to two main physical effects: line tension, which is caused by curvature of the step edge, and step-step interactions which can arise because of thermal step fluctuations, or elastic effects. This thesis focuses on axisymmetric crystals, with the result that the position of a step is uniquely described by a single scalar variable, and the step positions obey a coupled system of "step-flow" Ordinary Differential Equations (step flow ODEs). Chapter 2 of this thesis concentrates on the derivation and numerical solution of these equations, and their properties in the limits of slow adatom terrace diffusion and slow adatom attachment-detachment. Chapter 3 focuses on the analysis carried out by Margetis, Aziz and Stone ('MAS') [78] on a Partial Differential Equation (PDE) description of surface evolution.(cont.) Here, the crystal is also axisymmetric and has a single macroscopically flat region, a facet. It is discovered that the boundary condition of Step Chemical Potential Continuity, first suggested by Spohn [109] yields results that are inconsistent with the scalings predicted by the MAS analysis and with results from the step flow ODEs. The 'step drop' condition suggested by Israeli and Kandel [50] is implemented instead, and is shown to give good agreement with the results from the step flow ODEs. Chapters 4 and 5 explore the evolution of algebraic profiles: instead of starting with steps that are equally spaced, the step radii are initialized as a more general algebraic function of the height. In these two chapters, results are presented which involve approximate self-similarity of the profiles, a stability analysis of small perturbations, and quantification of decay rates. Chapter 6 of this thesis details the numerical procedure used to integrate the step flow equations. A 'multi-adaptive' time integrator is used where different time steps are taken for different components of the solution. This procedure has benefits over a standard integrator, because when a few steps cluster tightly together, these steps (and these steps only) become very stiff to integrate.(cont.) Whereas the inner most steps in the structure undergo a rapid motion, the majority of steps which are sufficiently far away from the facet, move relatively slowly and exhibit smooth behaviour in time. Using the same time step for all components in the solution is therefore quite inefficient. This chapter discusses the concept of "local stiffness", and how the motion of the inner most steps is handled.by Pak-Wing Fok.Ph.D

Hydromagnetic stability of the plasmapause.

Massachusetts Institute of Technology. Dept. of Earth and Planetary Sciences Thesis. 1970. Ph.D.Microfiche copy also available in Science Library.Vita.Bibliography: leaves 201-206.Ph.D

Studying Turbulence Using Numerical Simulation Databases

The Seventh Summer Program of the Center for Turbulence Research took place in the four-week period, July 5 to July 31, 1998. This was the largest CTR Summer Program to date, involving thirty-six participants from the U. S. and nine other countries. Thirty-one Stanford and NASA-Ames staff members facilitated and contributed to most of the Summer projects. A new feature, and perhaps a preview of the future programs, was that many of the projects were executed on non-NASA computers. These included supercomputers located in Europe as well as those operated by the Departments of Defense and Energy in the United States. In addition, several simulation programs developed by the visiting participants at their home institutions were used. Another new feature was the prevalence of lap-top personal computers which were used by several participants to carry out some of the work that in the past were performed on desk-top workstations. We expect these trends to continue as computing power is enhanced and as more researchers (many of whom CTR alumni) use numerical simulations to study turbulent flows. CTR's main role continues to be in providing a forum for the study of turbulence for engineering analysis and in facilitating intellectual exchange among the leading researchers in the field. Once again the combustion group was the largest. Turbulent combustion has enjoyed remarkable progress in using simulations to address increasingly complex and practically more relevant questions. The combustion group's studies included such challenging topics as fuel evaporation, soot chemistry, and thermonuclear reactions. The latter study was one of three projects related to the Department of Energy's ASCI Program (www.llnl.gov/asci); the other two (rocket propulsion and fire safety) were carried out in the turbulence modeling group. The flow control and acoustics group demonstrated a successful application of the so-called evolution algorithms which actually led to a previously unknown forcing strategy for jets yielding increased spreading rate. A very efficient algorithm for flow in complex geometries with moving boundaries based on the immersed boundary forcing technique was tested with very encouraging results. Also a new strategy for the destruction of aircraft trailing vortices was introduced and tested. The Reynolds Averaged Modeling (RANS) group demonstrated that the elliptic relaxation concept for RANS calculations is also applicable to transonic flows with shocks; however, prediction of laminar/turbulent transition remains an important pacing item. A large fraction of the LES effort was devoted to the development and testing of a new algorithmic procedure (as opposed to phenomenological model) for subgrid scale modeling based on regularized de-filtering of the flow variables. This appears to be a very promising approach, and a significant effort is currently underway to assess its robustness in high Reynolds number flows and in conjunction with numerical methods for complex flows. As part of the Summer Program two review tutorials were given on Turbulent structures in hydrocarbon pool fires (Sheldon Tieszen), and Turbulent combustion modeling: from RANS to LES via DNS (Luc Vervisch); and two seminars entitled Assessment of turbulence models for engineering applications (Paul Durbin) and Subgrid-scale modeling for non-premixed, turbulent reacting flows (James Riley) were presented. A number of colleagues from universities, government agencies, and industry attended the final presentations of the participants on July 31 and participated in the discussions. There are twenty-six papers in this volume grouped in five areas. Each group is preceded with an overview by its coordinator

Numerical simulation of the nonlinear response of composite plates under combined thermal and acoustic loading

A time-domain study of the random response of a laminated plate subjected to combined acoustic and thermal loads is carried out. The features of this problem also include given uniform static inplane forces. The formulation takes into consideration a possible initial imperfection in the flatness of the plate. High decibel sound pressure levels along with high thermal gradients across thickness drive the plate response into nonlinear regimes. This calls for the analysis to use von Karman large deflection strain-displacement relationships. A finite element model that combines the von Karman strains with the first-order shear deformation plate theory is developed. The development of the analytical model can accommodate an anisotropic composite laminate built up of uniformly thick layers of orthotropic, linearly elastic laminae. The global system of finite element equations is then reduced to a modal system of equations. Numerical simulation using a single-step algorithm in the time-domain is then carried out to solve for the modal coordinates. Nonlinear algebraic equations within each time-step are solved by the Newton-Raphson method. The random gaussian filtered white noise load is generated using Monte Carlo simulation. The acoustic pressure distribution over the plate is capable of accounting for a grazing incidence wavefront. Numerical results are presented to study a variety of cases

12th International Conference on Vibrations in Rotating Machinery

Since 1976, the Vibrations in Rotating Machinery conferences have successfully brought industry and academia together to advance state-of-the-art research in dynamics of rotating machinery. 12th International Conference on Vibrations in Rotating Machinery contains contributions presented at the 12th edition of the conference, from industrial and academic experts from different countries. The book discusses the challenges in rotor-dynamics, rub, whirl, instability and more. The topics addressed include: - Active, smart vibration control - Rotor balancing, dynamics, and smart rotors - Bearings and seals - Noise vibration and harshness - Active and passive damping - Applications: wind turbines, steam turbines, gas turbines, compressors - Joints and couplings - Challenging performance boundaries of rotating machines - High power density machines - Electrical machines for aerospace - Management of extreme events - Active machines - Electric supercharging - Blades and bladed assemblies (forced response, flutter, mistuning) - Fault detection and condition monitoring - Rub, whirl and instability - Torsional vibration Providing the latest research and useful guidance, 12th International Conference on Vibrations in Rotating Machinery aims at those from industry or academia that are involved in transport, power, process, medical engineering, manufacturing or construction

Annual research briefs, 1993

The 1993 annual progress reports of the Research Fellow and students of the Center for Turbulence Research are included. The first group of reports are directed towards the theory and application of active control in turbulent flows including the development of a systematic mathematical procedure based on the Navier Stokes equations for flow control. The second group of reports are concerned with the prediction of turbulent flows. The remaining articles are devoted to turbulent reacting flows, turbulence physics, experiments, and simulations

12th International Conference on Vibrations in Rotating Machinery

Since 1976, the Vibrations in Rotating Machinery conferences have successfully brought industry and academia together to advance state-of-the-art research in dynamics of rotating machinery. 12th International Conference on Vibrations in Rotating Machinery contains contributions presented at the 12th edition of the conference, from industrial and academic experts from different countries. The book discusses the challenges in rotor-dynamics, rub, whirl, instability and more. The topics addressed include: - Active, smart vibration control - Rotor balancing, dynamics, and smart rotors - Bearings and seals - Noise vibration and harshness - Active and passive damping - Applications: wind turbines, steam turbines, gas turbines, compressors - Joints and couplings - Challenging performance boundaries of rotating machines - High power density machines - Electrical machines for aerospace - Management of extreme events - Active machines - Electric supercharging - Blades and bladed assemblies (forced response, flutter, mistuning) - Fault detection and condition monitoring - Rub, whirl and instability - Torsional vibration Providing the latest research and useful guidance, 12th International Conference on Vibrations in Rotating Machinery aims at those from industry or academia that are involved in transport, power, process, medical engineering, manufacturing or construction