On the Chebyshev collocation spectral approach to stability of fluid flow in a porous medium

In this paper, the temporal development of small disturbances in a pressure‐driven fluid flow through a channel filled with a saturated porous medium is investigated. The Brinkman flow model is employed in order to obtain the basic flow velocity distribution. Under normal mode assumption, the linearized governing equations for disturbances yield a fourth‐order eigenvalue problem, which reduces to the well‐known Orr–Sommerfeld equation in some limiting cases solved numerically by a spectral collocation technique with expansions in Chebyshev polynomials. The critical Reynolds number Re_c, the critical wave number α_c, and the critical wave speed c_c are obtained for a wide range of the porous medium shape factor parameter S. It is found that a decrease in porous medium permeability has a stabilizing effect on the fluid flow. Copyright © 2008 John Wiley & Sons, Ltd.     

A new difference scheme with high accuracy and absolute stability for solving convection–diffusion equations

In this paper, we use a semi-discrete and a padé approximation method to propose a new difference scheme for solving convection–diffusion problems. The truncation error of the difference scheme is O(h⁴+τ⁵). It is shown through analysis that the scheme is unconditionally stable. Numerical experiments are conducted to test its high accuracy and to compare it with Crank–Nicolson method.     

Two new examples of sets without medians and centers

Many problems in continuous location theory, reduce to finding a best location, in the sense that a facility must be located at a point minimizing the sum of distances to the points of a given finite set (median) or the largest distances to all points (center). The setting is often assumed to be a Banach space. To have a better understanding concerning the structure of location problems, it may occur also in rather simple cases. In this paper we indicate two simple examples of four-point sets such that one of the two problems indicated has a solution, while the other one has no solution. Also, we list papers containing examples previously given, dealing with this lack of optimal solutions.     

An efficient transient Navier–Stokes solver on compact nonuniform space grids

In this paper, we propose an implicit higher-order compact (HOC) finite difference scheme for solving the two-dimensional (2D) unsteady Navier–Stokes (N–S) equations on nonuniform space grids. This temporally second-order accurate scheme which requires no transformation from the physical to the computational plane is at least third-order accurate in space, which has been demonstrated with numerical experiments. It efficiently captures both transient and steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. The proposed scheme is likely to be very useful for the computation of transient viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variation. Numerical results are presented and compared with analytical as well as established numerical data. Excellent comparison is obtained in all the cases.     

Fast Face-Recognition Optical Parallel Correlator Using High Accuracy Correlation Filter

We designed and fabricated a fully automatic fast face recognition optical parallel correlator [E. Watanabe and K. Kodate: Appl. Opt. 44 (2005) 5666] based on the VanderLugt principle. The implementation of an as-yet unattained ultra high-speed system was aided by reconfiguring the system to make it suitable for easier parallel processing, as well as by composing a higher accuracy correlation filter and high-speed ferroelectric liquid crystal-spatial light modulator (FLC-SLM). In running trial experiments using this system (dubbed FARCO), we succeeded in acquiring remarkably low error rates of 1.3% for false match rate (FMR) and 2.6% for false non-match rate (FNMR). Given the results of our experiments, the aim of this paper is to examine methods of designing correlation filters and arranging database image arrays for even faster parallel correlation, underlining the issues of calculation technique, quantization bit rate, pixel size and shift from optical axis. The correlation filter has proved its excellent performance and higher precision than classical correlation and joint transform correlator (JTC). Moreover, arrangement of multi-object reference images leads to 10-channel correlation signals, as sharply marked as those of a single channel. This experiment result demonstrates great potential for achieving the process speed of 10000 face/s.     

Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part II: Clenshaw-Lord type approximants

Laurent-Padé (Chebyshev) rational approximantsP _m (w, w ⁻¹)/Q _n (w, w ⁻¹) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP _m /Q _n matches that of a given functionf(w, w ⁻¹) up to terms of orderw ^±(m+n) , based only on knowledge of the Laurent series coefficients off up to terms inw ^±(m+n) . This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent series ofP _m matches that ofQ _n f up to terms of orderw ^±(m+n ), but based on knowledge of the series coefficients off up to terms inw ^±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the use of either.     

Numerical Approximation of the Spectra of Phan-Thien Tanner Liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear PTT fluid is stable to infinitesimal disturbances with respect to the range of shear-thinning, extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.     

A new class of three-variable orthogonal polynomials and their recurrences relations

A new class of three-variable orthogonai polynomials,defined as eigenfunctions of a second order PDE operator,is studied.These polynomials are orthogonal over a curved tetrahedron region, which can be seen as a mapping from a traditional tetrahedron,and can be taken as an extension of the 2-D Steiner domain.The polynomials can be viewed as Jacobi polynomials on such a domain.Three- term relations are derived explicitly.The number of the individual terms,involved in the recurrences relations,are shown to be independent on the total degree of the polynomials.The numbers now are determined to be five and seven,with respect to two conjugate variables z,(?) and a real variable r, respectively.Three examples are discussed in details,which can be regarded as the analogues of the Chebyshev polynomials of the first and the second kinds,and Legendre polynomials.     

The Use of Multi-Pass Time-of-Flight Mass Analyzers for Nuclear-Decay Spectroscopy of Mass Identified Nuclei

The masses of ions of some keV can be determined in multi-pass time-of-flight mass analyzers [1,2] with high precision. The mass accuracies thus achieved are sufficient to determine the proton and neutron numbers for most short-lived and stable nuclei [3,5]. Recording α- or γ-radiation of the investigated nuclei in delayed coincidence to the ion arrival, one thus can perform nuclear spectroscopy of selected nuclei. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chebyshev pseudospectral method for computing numerical solution of convection–diffusion equation

A method for computing highly accurate numerical solutions of 1D convection–diffusion equations is proposed. In this method, the equation is first discretized with respect to the spatial variable, transforming the original problem into a set of ordinary differential equations, and then the resulting system is integrated in time by the fourth-order Runge–Kutta method. Spatial discretization is done by using the Chebyshev pseudospectral collocation method. Before describing the method, we review a finite difference-based method by Salkuyeh [D. Khojasteh Salkuyeh, On the finite difference approximation to the convection–diffusion equation, Appl. Math. Comput. 179 (2006) 79–86], and, contrary to the proposal of the author, we show that this method is not suitable for problems involving time dependent boundary conditions, which calls for revision. Stability analysis based on pseudoeigenvalues to determine the maximum time step for the proposed method is also carried out. Superiority of the proposed method over a revised version of Salkuyeh’s method is verified by numerical examples.