A multidomain spectral collocation method for the Stokes problem

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

On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function

It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed

On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

AbstractIt is well known that the Fourier series of an analytic and periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm. It is also known that if the function is not periodic, the rate of convergence deteriorates; in particular, there is no convergence in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. In this paper we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed. The proof is a constructive one and makes use of the Gegenbauer polynomials Cλn(x). It consists of two steps. In the first step we show that the first m coefficients of the Gegenbauer expansion (based on Cλn(x), for 0⩽n⩽m) of any L2 function can be obtained, within exponential accuracy, provided that both λ and m are proportional to (but smaller than) N. In the second step we construct the Gegenbauer expansion based on Cλn, 0⩽n⩽m, from the coefficients found in the first step. We show that this series converges exponentially with N, provided that the original function is analytic (though nonperiodic). Thus we prove that the Gibbs phenomenon can be completely overcome

A spectral collocation approximation for the radial-infall of a compact object into a Schwarzschild black hole

The inhomogeneous Zerilli equation is solved in time-domain numerically with the Chebyshev spectral collocation method to investigate a radial-infall of the point particle towards a Schwarzschild black hole. Singular source terms due to the point particle appear in the equation in the form of the Dirac $\delta$-function and its derivative. For the approximation of singular source terms, we use the direct derivative projection method without any regularization. The gravitational waveforms are evaluated as a function of time. We compare the results of the spectral collocation method with those of the explicit second-order central-difference method. The numerical results show that the spectral collocation approximation with the direct projection method is accurate and converges rapidly when compared with the finite-difference method.Comment: Accepted for publication in International Journal of Modern Physics C (IJMPC

Software Acquisition and Project Management (KSU)

This Grants Collection for Software Acquisition and Project Management was created under a Round Eleven ALG Textbook Transformation Grant. Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process. Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials: Linked Syllabus Initial Proposal Final Reporthttps://oer.galileo.usg.edu/compsci-collections/1033/thumbnail.jp

Management of IT (KSU)

This Grants Collection for Management of IT was created under a Round Eleven ALG Textbook Transformation Grant. Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process. Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials: Linked Syllabus Initial Proposal Final Reporthttps://oer.galileo.usg.edu/compsci-collections/1027/thumbnail.jp

Hardware and Software (KSU)

This Grants Collection for Hardware and Software was created under a Round Eleven ALG Textbook Transformation Grant. Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process. Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials: Linked Syllabus Initial Proposal Final Reporthttps://oer.galileo.usg.edu/compsci-collections/1026/thumbnail.jp

Aeroacoustic Study of a High-Fidelity Aircraft Model: Part 1- Steady Aerodynamic Measurements

In this paper, we present steady aerodynamic measurements for an 18% scale model of a Gulfstream air-craft. The high fidelity and highly-instrumented semi-span model was developed to perform detailed aeroacoustic studies of airframe noise associated with main landing gear/flap components and gear-flap interaction noise, as well as to evaluate novel noise reduction concepts. The aeroacoustic tests, being conducted in the NASA Langley Research Center 14- by 22-Foot Subsonic Tunnel, are split into two entries. The first entry, completed November 2010, was entirely devoted to the detailed mapping of the aerodynamic characteristics of the fabricated model. Flap deflections of 39?, 20?, and 0? with the main landing gear on and off were tested at Mach numbers of 0.16, 0.20, and 0.24. Additionally, for each flap deflection, the model was tested with the tunnel both in the closed-wall and open-wall (jet) modes. During this first entry, global forces (lift and drag) and extensive steady and unsteady surface pressure measurements were obtained. Preliminary analysis of the measured forces indicates that lift, drag, and stall characteristics compare favorably with Gulfstream?s high Reynolds number flight data. The favorable comparison between wind-tunnel and flight data allows the semi-span model to be used as a test bed for developing/evaluating airframe noise reduction concepts under a relevant environment. Moreover, initial comparison of the aerodynamic measurements obtained with the tunnel in the closed- and open-wall configurations shows similar aerodynamic behavior. This permits the acoustic and off-surface flow measurements, planned for the second entry, to be conducted with the tunnel in the open-jet mode