Compact finite difference - Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported.

Direct numerical simulation of autoignition in a non-premixed, turbulent medium

In this paper, direct numerical simulation of autoignition in an initially non-premixed medium under isotropic, homogeneous, and decaying turbulence is presented. The pressure-based method developed herein is a spectral implementation of the sequential steps followed in the predictor-corrector type of algorithms; it includes the effects of density fluctuations caused by spatial inhomogeneities ill temperature and species. The velocity and pressure field are solved in the spectral space while the scalars and density field are solved in the physical space. The presented results reveal that the autoignition spots originate and evolve at locations where (1) the composition corresponds to a small range around a specific mixture fraction, and (2) the conditional scaler dissipation rate is low. A careful examination of the data obtained indicates that the autoignition spots originate in the vortex cores, and the hot gases travel outward as combustion progresses. Hence, the applicability of the transient laminar flamelet model for this problem is questioned. The dependence of autoignition characteristics on parameters such as (1) die initial eddy-turnover time and (2) the initial ratio of length scale of scalars to that of velocities are investigated. Certain implications of new results on the conditional moment closure modeling are discussed.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

Bathymetric map of Heron Reef, Australia, derived from airborne hyperspectral data at 1 m resolution

A simple method for efficient inversion of arbitrary radiative transfer models for image analysis is presented. The method operates by representing the shape of the function that maps model parameters to spectral reflectance by an adaptive look-up tree (ALUT) that evenly distributes the discretization error of tabulated reflectances in spectral space. A post-processing step organizes the data into a binary space partitioning tree that facilitates an efficient inversion search algorithm. In an example shallow water remote sensing application, the method performs faster than an implementation of previously published methodology and has the same accuracy in bathymetric retrievals. The method has no user configuration parameters requiring expert knowledge and minimizes the number of forward model runs required, making it highly suitable for routine operational implementation of image analysis methods. For the research community, straightforward and robust inversion allows research to focus on improving the radiative transfer models themselves without the added complication of devising an inversion strategy.

Kneser and Hereditarily Kneser Subgroups of a Profinite Group

2000 Mathematics Subject Classification: 20E18, 12G05, 12F10, 12F99.

Towards automatic tree crown detection and delineation in spectral feature space using PCNN and morphological reconstruction

The application of object-based approaches to the problem of extracting vegetation information from images requires accurate delineation of individual tree crowns. This paper presents an automated method for individual tree crown detection and delineation by applying a simplified PCNN model in spectral feature space followed by post-processing using morphological reconstruction. The algorithm was tested on high resolution multi-spectral aerial images and the results are compared with two existing image segmentation algorithms. The results demonstrate that our algorithm outperforms the other two solutions with the average accuracy of 81.8%.

A Crank--Nicolson ADI spectral method for a two-dimensional riesz space fractional nonlinear reaction-diffusion equation

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order $2$ in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains ofRn. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

Space-resolved spectral diagnosis of line-shaped laser-produced magnesium plasmas

Space-resolved spectra of line-shaped laser-produced magnesium plasmas in the normal direction of the target have been obtained using a pinhole crystal spectrograph. These spectra are treated by a spectrum analyzing code for obtaining the true spectra and fine structures of overlapped lines. The spatial distributions of electron temperature and density along the normal direction of the target surface have been obtained with different spectral diagnostic techniques. Especially, the electron density plateaus beyond the critical surface in line-shaped magnesium plasmas have been obtained with a fitting technique applied to the Stark-broadened Ly-alpha wings of hydrogenic ions. The difference of plasma parameters between those obtained by different diagnostic techniques is discussed. Other phenomena, such as plasma satellites, population inversion, etc., which are observed in magnesium plasmas, are also presented.

Can we measure terrestrial photosynthesis from space directly, using spectral reflectance and fluorescence?

Attempts to estimate photosynthetic rate or gross primary productivity from remotely sensed absorbed solar radiation depend on knowledge of the light use efficiency (LUE). Early models assumed LUE to be constant, but now most researchers try to adjust it for variations in temperature and moisture stress. However, more exact methods are now required. Hyperspectral remote sensing offers the possibility of sensing the changes in the xanthophyll cycle, which is closely coupled to photosynthesis. Several studies have shown that an index (the photochemical reflectance index) based on the reflectance at 531 nm is strongly correlated with the LUE over hours, days and months. A second hyperspectral approach relies on the remote detection of fluorescence, which is a directly related to the efficiency of photosynthesis. We discuss the state of the art of the two approaches. Both have been demonstrated to be effective, but we specify seven conditions required before the methods can become operational.

Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1

Spectral theory of operators in Hilbert space. [Lecture notes]

Bibliography: leaf [205]

Motivation, development and validation of a new spectral greenness index : a spectral dimension related to foliage projective cover

A method is presented for the development of a regional Landsat-5 Thematic Mapper (TM) and Landsat-7 Enhanced Thematic Mapper plus (ETM+) spectral greenness index, coherent with a six-dimensional index set, based on a single ETM+ spectral image of a reference landscape. The first three indices of the set are determined by a polar transformation of the first three principal components of the reference image and relate to scene brightness, percent foliage projective cover (FPC) and water related features. The remaining three principal components, of diminishing significance with respect to the reference image, complete the set. The reference landscape, a 2200 km2 area containing a mix of cattle pasture, native woodland and forest, is located near Injune in South East Queensland, Australia. The indices developed from the reference image were tested using TM spectral images from 19 regionally dispersed areas in Queensland, representative of dissimilar landscapes containing woody vegetation ranging from tall closed forest to low open woodland. Examples of image transformations and two-dimensional feature space plots are used to demonstrate image interpretations related to the first three indices. Coherent, sensible, interpretations of landscape features in images composed of the first three indices can be made in terms of brightness (red), foliage cover (green) and water (blue). A limited comparison is made with similar existing indices. The proposed greenness index was found to be very strongly related to FPC and insensitive to smoke. A novel Bayesian, bounded space, modelling method, was used to validate the greenness index as a good predictor of FPC. Airborne LiDAR (Light Detection and Ranging) estimates of FPC along transects of the 19 sites provided the training and validation data. Other spectral indices from the set were found to be useful as model covariates that could improve FPC predictions. They act to adjust the greenness/FPC relationship to suit different spectral backgrounds. The inclusion of an external meteorological covariate showed that further improvements to regional-scale predictions of FPC could be gained over those based on spectral indices alone.