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An explicit time-advancing scheme for the spectral solution of parabolic equations is presented. Several two-dimensional examples are considered, including convection-diffusion and nonlinear problems, under various boundary conditions. Numerical evidence demonstrates the efficiency and accuracy of the spectral approach.
We implement a general relativistic neutrino transport scheme appropriate to problems in stellar collapse and neutron star formation. In this scheme the spectral energy and momentum balance equations are solved as a coupled set. We perform transport calculations on two frozen stellar background models neglecting the feedback of the neutrinos on the matter. We compare with schemes that ignore either relativity or momentum balance or both. The work performed by the neutrino stress on the matter is taken into account and has an important effect. In particular, if there is a hydrodynamical shock in the system, this work contribution boosts the energy deposition behind the shock. This may enhance the chances for a successful neutrino driven supernova explosion.
The Apollo Gamma Ray Spectrometer which was part of the scientific instrument module aboard Apollo 16 obtained more than 20 hours of astronomical data near the end of the transearth coast phase of the mission. During this time the vehicle was maneuvered so that every part of the sky as seen by the spectrometer was eventually occulted by the spacecraft, providing a valuable data base for gamma ray astronomy in the 100 keV to 5 MeV spectral range. Schemes classified according to the underlying assumed source distribution are analyzed. It is known that there is at least one point source, the Crab, and it can be seen in the data. Methods for data analysis are shown. In one method looking only for point sources, some preliminary results concerning the Crab nebula are presented which confirm previous measurement of the strength of that source.
A spectral multigrid scheme is described which can solve pseudospectral discretizations of self-adjoint elliptic problems in O(N log N) operations. An iterative technique for efficiently implementing semi-implicit time-stepping for pseudospectral discretizations of Navier-Stokes equations is discussed. This approach can handle variable coefficient terms in an effective manner. Pseudospectral solutions of compressible flow problems are presented. These include one dimensional problems and two dimensional Euler solutions. Results are given both for shock-capturing approaches and for shock-fitting ones.
The effects on calculated lower-level population densities of the truncation of Na and Sr(+) model atoms are determined in the context of the present spectral diagnostic scheme for solar prominences and spicules. It is shown that neglect of the upper atomic levels in Na, in particular, leads to overestimates in electron density and gas pressure by factors of about 2 and about 4, respectively, and to underestimates in the degree of hydrogen ionization and in the line-of-sight thickness of emitting material again by factors of about 2 and about 4, respectively. The implications of the revised emitting region extents, in particular, on the validity of the diagnostic method for these features are discussed.
A multidomain spectral collocation scheme is proposed for the approximation of the two-dimensional Stokes problem. It is shown that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.
Proposed keratometer based on scheme involving spectral dispersal of white-light interference fringes. Instrument operates in "snapshot" mode: no scanning necessary, not necessary to immobilize patient's eye. Insensitive to vibration, involves no phase shifting, and has variable sensitivity. Intended primarily for use in medical assessments of human corneas, also used to measure shapes of animal corneas, lenses, and other aspherical or spherical reflective or partly reflective surfaces.
A fully nonlinear 3-dimensional numerical model (GEOSIM), previously developed and validated for several cases of geophysical fluid flow, has been used to investigate the dynamical behavior of laboratory experiments of fluid flows similar to those of the Earth's atmosphere. The phenomena investigated are amplitude vacillation, and the response of the fluid system to uneven heating and cooling. The previous year's work included hysteresis in the transition between axisymmetric and wave flow. Investigation is also continuing of the flows in the Geophysical Fluid Flow Cell (GFFC), a low-gravity Spacelab experiment. Much of the effort in the past year has been spent in validation of the model under a wide range of external parameters including nonlinear flow regimes. With the implementation of a 3-dimensional upwind differencing scheme, higher spectral resolution, and a shorter time step, the model has been found capable of predicting the majority of flow regimes observed in one complete series of baroclinic annulus experiments of Pfeffer and co-workers. Detailed analysis of amplitude vacillation has revealed that the phase splitting described in the laboratory experiments occurs in some but not all cases. Through the use of animation of the models output, a vivid 3-dimensional view of the phase splitting was shown to the audience of the Southeastern Geophysical Fluid Dynamics Conference in March of this year. A study on interannual variability was made using GEOSIM with periodic variations in the thermal forcing. Thus far, the model has not predicted a chaotic behavior as observed in the experiments, although there is a sensitivity in the wavenumber selection to the initial conditions. Work on this subject, and on annulus experiments with non-axisymmetric thermal heating, will continue. The comparison of GEOSIM's predictions will result from the Spacelab 3 GFFC experiments continued over the past year, on a 'back-burner' basis. At this point, the study (in the form of a draft of a journal article) is nearly completed. The results from GEOSIM compared very well with the experiments, and the use of the model allows the demonstration of flow mechanics that were not possible with the experimental data. For example, animation of the model output shows that the forking of the spiral bands is a transient phenomenon, due to the differential east-west propagation of convection bands from different latitudes.
An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.
The effect of reducing the formal order of accuracy of a finite-difference scheme in order to optimize its high-frequency performance is investigated using the I-D nonlinear unsteady inviscid Burgers'equation. It is found that the benefits of optimization do carry over into nonlinear applications. Both explicit and compact schemes are compared to Tam and Webb's explicit 7-point Dispersion Relation Preserving scheme as well as a Spectral-like compact scheme derived following Lele's work. Results are given for the absolute and L2 errors as a function of time.
Human transformation of the terrestrial biosphere via resource utilization is a critical impetus for monitoring and characterizing anthropogenic change to vegetation condition. The primary objective of this research was to detect anthropogenic forest disturbance for a recent Landsat time series. A novel combination of an autonomous change detection procedure and spectral classification scheme was applied and tested in a landscape that has undergone significant resource development over the last 30 years. Anthropogenic disturbance was detected with greater than 93% accuracy. Most disturbances were correctly classified as within +/-1 year. The signal of anthropogenic disturbance was significant in the landscape, accounting for more than 91% of all disturbances and 86% of total disturbed area during the 23-year study period. The study demonstrated a robust approach for examining historical disturbance trends related to human-modification of the environment
Cloud microphysics is inevitably affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effects of atmospheric aerosol concentration on cloud development, r d a U production, and rainfall rates for convective clouds. Recently, two detailed spectral-bin microphysical schemes were implemented into the Goddard Cumulus Ensembe1 (GCE) model. The formulation for the explicit spectral-bin microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e. pristine ice crystals (columnar and platelike), snow (dendrites and aggregates), graupel and frozen drops/hail]. Each type is described by a special size distribution function containing many categories (i.e. 33 bins). Atmospheric aerosols are also described using number density size-distribution functions. A spectral-bin microphysical model is very expensive from a computational point of view and has only been implemented into the 2D version of the GCE at the present time. The model is tested by studying the evolution of deep tropical clouds in the west Pacific warm pool region and in the mid-latitude continent with different concentrations of CCN: a low "c1ean"concentration and a high "dirty" concentration. In addition, differences and similarities between bulk microphysics and spectral-bin microphysical schemes will be examined and discussed.
Cloud microphysics is inevitably affected by the smoke particle (CCN, cloud condensation nuclei) size distributions below the clouds. Therefore, size distributions parameterized as spectral bin microphysics are needed to explicitly study the effects of atmospheric aerosol concentration on cloud development, rainfall production, and rainfall rates for convective clouds. Recently, two detailed spectral-bin microphysical schemes were implemented into the Goddard Cumulus Ensembel (GCE) model. The formulation for the explicit spectral-bin microphysical processes is based on solving stochastic kinetic equations for the size distribution functions of water droplets (i.e., cloud droplets and raindrops), and several types of ice particles [i.e. pristine ice crystals (columnar and plate-like), snow (dendrites and aggregates), graupel and frozen drops/hail]. Each type is described by a special size distribution function containing many categories (i.e. 33 bins). Atmospheric aerosols are also described using number density size distribution functions. A spectral-bin microphysical model is very expensive from a computational point of view and has only been implemented into the 2D version of the GCE at the present time. The model is tested by studying the evolution of deep tropical clouds in the west Pacific warm pool region and in the mid-latitude continent with different concentrations of CCN: a low "c1ean"concentration and a high "dirty" concentration. In addition, differences and similarities between bulk microphysics and spectral-bin microphysical schemes will be examined and discussed.
A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.
The 'hotspots' of powerful radio galaxies (the compact, high brightness regions, where the jet flow collides with the intergalactic medium (IGM)) have been imaged in radio, optical and recently in X-ray frequencies. We propose a scheme that unifies their, at first sight, disparate broad band (radio to X-ray) spectral properties. This scheme involves a relativistic flow upstream of the hotspot that decelerates to the sub-relativistic speed of its inferred advance through the IGM and it is viewed at different angles to its direction of motion, as suggested by two independent orientation estimators (the presence or not of broad emission lines in their optical spectra and the core-to-extended radio luminosity). This scheme, besides providing an account of the hotspot spectral properties with jet orientation, it also suggests that the large-scale jets remain relativistic all the way to the hotspots.
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A suitable theoretical scheme able to describe, in a wide variety of astrophysical situations, the phenomenon of atomic line polarization by electron impact is developed. Starting from the general principles of quantum mechanics and assuming the Born approximation, the rate equations for the density matrix elements of a multilevel atomic system, interacting with a nonrelativistic electron beam having any kind of angular distribution, are derived in full generality. The resulting theory generalizes the previous ones by accounting for the collisional rates and the cross sections concerning both inelastic and superelastic collisions (in any geometrical situation), and, moreover, by taking into account the coherences among Zeeman sublevels split by a magnetic field. As an example of particular relevance, the general formulas derived in the first sections of the paper are subsequently particularized to the case of the electric dipole interaction.
In a series of papers, Olsson (1994, 1995), Olsson & Oliger (1994), Strand (1994), Gerritsen Olsson (1996), Yee et al. (1999a,b, 2000) and Sandham & Yee (2000), the issue of nonlinear stability of the compressible Euler and Navier-Stokes Equations, including physical boundaries, and the corresponding development of the discrete analogue of nonlinear stable high order schemes, including boundary schemes, were developed, extended and evaluated for various fluid flows. High order here refers to spatial schemes that are essentially fourth-order or higher away from shock and shear regions. The objective of this paper is to give an overview of the progress of the low dissipative high order shock-capturing schemes proposed by Yee et al. (1999a,b, 2000). This class of schemes consists of simple non-dissipative high order compact or non-compact central spatial differencings and adaptive nonlinear numerical dissipation operators to minimize the use of numerical dissipation. The amount of numerical dissipation is further minimized by applying the scheme to the entropy splitting form of the inviscid flux derivatives, and by rewriting the viscous terms to minimize odd-even decoupling before the application of the central scheme (Sandham & Yee). The efficiency and accuracy of these scheme are compared with spectral, TVD and fifth- order WENO schemes. A new approach of Sjogreen & Yee (2000) utilizing non-orthogonal multi-resolution wavelet basis functions as sensors to dynamically determine the appropriate amount of numerical dissipation to be added to the non-dissipative high order spatial scheme at each grid point will be discussed. Numerical experiments of long time integration of smooth flows, shock-turbulence interactions, direct numerical simulations of a 3-D compressible turbulent plane channel flow, and various mixing layer problems indicate that these schemes are especially suitable for practical complex problems in nonlinear aeroacoustics, rotorcraft dynamics, direct numerical simulation or large eddy simulation of compressible turbulent flows at various speeds including high-speed shock-turbulence interactions, and general long time wave propagation problems. These schemes, including entropy splitting, have also been extended to freestream preserving schemes on curvilinear moving grids for a thermally perfect gas (Vinokur & Yee 2000).