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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

Iterative design of one- and two-dimensional FIR digital filters

The paper describes a new iterative technique for designing FIR (finite duration impulse response) digital filters using a frequency weighted least squares approximation. The technique is as easy to implement (via FFT) and as effective in two dimensions as in one dimension, and there are virtually no limitations on the class of filter frequency spectra approximated. An adaptive adjustment of the frequency weight to achieve other types of design approximation such as Chebyshev type design is discussed.

Suk, M.↗
NASA NTRS1979

Accurate numerical evaluation of modified Struve functions occurring in unsteady aerodynamics

A method is developed for computing the modified Struve functions that occur in unsteady aerodynamics. The method uses a rational approximation supplemented by an asymptotic series for large argument. Simple recursive formulas for generating the coefficients are derived. The method is capable of generating results of arbitrary accuracy. It can also be used for complex argument and order. For greater computing speed, a method is presented that uses the rational and asymptotic approximations to generate Chebyshev coefficients.

Desmarais, R. N.↗
NASA NTRS1989

Chebyshev Polynomials Are Not Always Optimal

The authors are concerned with the problem of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on Epsilon. Here, Epsilon is a given ellipse with both foci on the real axis and c is a given real point not contained in Epsilon. Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this note, the authors show that this is not true in general. Moreover, the authors derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.

CHEBYSHEV APPROXIMATION↗
NASA NTRS1975

Compression of ephemerides

An algorithm is proposed for generating sequences of Chebyshev series which are the best approximations of an astronomical ephemeris in the sense of Chebyshev over large intervals of time. The criterion for a polynomial approximation of a function to be the best polynomial approximation of the function is that the error function present certain rippling characteristics as described by Remez (1957). General features of the program in PL/1 are described.

Deprit, A.↗
NASA NTRS1990

Quadrature imposition of compatibility conditions in Chebyshev methods

Often, in solving an elliptic equation with Neumann boundary conditions, a compatibility condition has to be imposed for well-posedness. This condition involves integrals of the forcing function. When pseudospectral Chebyshev methods are used to discretize the partial differential equation, these integrals have to be approximated by an appropriate quadrature formula. The Gauss-Chebyshev (or any variant of it, like the Gauss-Lobatto) formula can not be used here since the integrals under consideration do not include the weight function. A natural candidate to be used in approximating the integrals is the Clenshaw-Curtis formula, however it is shown that this is the wrong choice and it may lead to divergence if time dependent methods are used to march the solution to steady state. The correct quadrature formula is developed for these problems. This formula takes into account the degree of the polynomials involved. It is shown that this formula leads to a well conditioned Chebyshev approximation to the differential equations and that the compatibility condition is automatically satisfied.

Gottlieb, D.↗
NASA NTRS1991

A Fourier-Chebyshev pseudospectral method for solving steady 3-D Navier-Stokes and heat equations in cylindrical cavities

A Fourier-Chebyshev pseudospectral method for solving steady 3D Navier-Stokes equations in cylindrical cavities is presented and discussed. The general method is pseudo-unsteady and uses a semi-implicit finite difference scheme for the time discretization. The generalized ADI (Alternating Direction Implicit) procedure is then applied to reduce the problem to a successive solution of one-dimensional problems. The spatial approximation uses a Fourier-Galerkin approximation in the periodic direction and a Chebyshev-collocation approximation in the other directions. Difficulties related to the pressure are surmounted by using the artifical compressibility method. A suitable variable change was chosen to avoid the problem of singularity at the axis generated by cylindrical coordinates. The method is first tested on an advection-diffusion equation and then on the Navier-Stokes equations. Finally, the method is illustrated by a convection problem of a differentially heated fluid.

Pulicani, J. P.↗
NASA NTRS1994

Spectral solution of the viscous blunt body problem. 2: Multidomain approximation

We present steady solutions of high speed viscous flows over blunt bodies using a multidomain Chebyshev spectral collocation method. The region with the shock layer is divided into subdomains so that internal layers can be well-resolved. In the interiors of the subdomains, the solution is approximated by Chebyshev collocation. At interfaces between subdomains, the advective terms are upwinded and the viscous terms are treated by a penalty method. The method is applied to five flows, the Mach number range 5-25 and Reynolds number range 2,000 - 83,000, based on nose radius. Results are compared to experimental data and to a finite difference result.

Kopriva, David A.↗
NASA NTRS1985

Well-posedness of one-way wave equations and absorbing boundary conditions

A one-way wave equation is a partial differential which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of (1-s sup 2) on -1, 1 by a rational function r(s) = p sub m (s)/q sub n(s). Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m, n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation.

Trefethen, L. N.↗
NASA NTRS1989

Chebyshev polynomials are not always optimal

The problem is that of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on Epsilon. Here, Epsilon is a given ellipse with both foci on the real axis and c is a given real point not contained in Epsilon. Problems of this type arise in certain iterative matrix computations and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. It is shown that this is not true in general. Moreover, sufficient conditions are derived which guarantee that Chebyshev polynomials are optimal. Some numerical examples are also presented.

Fischer, Bernd↗
NASA NTRS1982

On the geodetic applications of simultaneous range-differencing to LAGEOS

The possibility of improving the accuracy of geodetic results by use of simultaneously observed ranges to Lageos, in a differencing mode, from pairs of stations was studied. Simulation tests show that model errors can be effectively minimized by simultaneous range differencing (SRD) for a rather broad class of network satellite pass configurations. The methods of least squares approximation are compared with monomials and Chebyshev polynomials and the cubic spline interpolation. Analysis of three types of orbital biases (radial, along- and across track) shows that radial biases are the ones most efficiently minimized in the SRC mode. The degree to which the other two can be minimized depends on the type of parameters under estimation and the geometry of the problem. Sensitivity analyses of the SRD observation show that for baseline length estimations the most useful data are those collected in a direction parallel to the baseline and at a low elevation. Estimating individual baseline lengths with respect to an assumed but fixed orbit not only decreases the cost, but it further reduces the effects of model biases on the results as opposed to a network solution. Analogous results and conclusions are obtained for the estimates of the coordinates of the pole.

Pablis, E. C.↗
NASA NTRS1987

An adaptive pseudo-spectral method for reaction diffusion problems

The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.

Bayliss, A.↗
NASA NTRS1989

An adaptive pseudo-spectral method for reaction diffusion problems

The spectral interpolation error was considered for both the Chebyshev pseudo-spectral and Galerkin approximations. A family of functionals I sub r (u), with the property that the maximum norm of the error is bounded by I sub r (u)/J sub r, where r is an integer and J is the degree of the polynomial approximation, was developed. These functionals are used in the adaptive procedure whereby the problem is dynamically transformed to minimize I sub r (u). The number of collocation points is then chosen to maintain a prescribed error bound. The method is illustrated by various examples from combustion problems in one and two dimensions.

Bayliss, A.↗
NASA NTRS1993

On the Gibbs phenomenon 3: Recovering exponential accuracy in a sub-interval from a spectral partial sum of a piecewise analytic function

The investigation of overcoming Gibbs phenomenon was continued, i.e., obtaining exponential accuracy at all points including at the discontinuities themselves, from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function. It was shown that if we are given the first N expansion coefficients of an L(sub 2) function f(x) in terms of either the trigonometrical polynomials or the Chebyshev or Legendre polynomials, an exponentially convergent approximation to the point values of f(x) in any sub-interval in which it is analytic can be constructed.

Gottlieb, David↗
NASA NTRS1976

Control by model error estimation

Modern control theory relies upon the fidelity of the mathematical model of the system. Truncated modes, external disturbances, and parameter errors in linear system models are corrected by augmenting to the original system of equations an 'error system' which is designed to approximate the effects of such model errors. A Chebyshev error system is developed for application to the Large Space Telescope (LST).

Likins, P. W.↗
NASA NTRS1990

Analysis of spectral projectors in one-dimensional domains

A class of projection operators with values in a subspace of polynomials is analyzed. These projection operators are related to the Hilbert spaces involved in the numerical analysis of spectral methods. They are, in the first part of the paper, the standard Sobolev spaces and, in the second part, some weighted Sobolev spaces, the weight of which is related to the orthoronality relation satisfied by the Chebyshev polynomials. These results are used to study the approximation of a model fourth-order problem.

Maday, Y.↗
NASA NTRS1986

Well-posedness of one-way wave equations and absorbing boundary conditions

The one-way wave equations occurring in geophysics, underwater acoustics, and numerical studies involving absorbing boundary conditions are characterized analytically. The conditions under which such equations are well posed are obtained by examining the rational functions used to reduce them. Results are presented for Chebyshev, Pade, least-squares, and symmetric least-squares approximations.

Trefethen, Lloyd N.↗
DOE OSTI2026

Charge And Dynamic Current On Tubular Antennas For Various Drive Conditions

The mixed boundary value problem of a tubular conductor is solved using an approximate representation of its Fourier coefficients. A two term solution is derived, which represents the solution over an extremely broad range of aspect ratios. This representation is used to find the electrostatic solution and capacitance of a charged tube as well as the solution of a tube in a uniform field and its dipole moment. This second case is directly useful as a model for a monopole electric field probe. This approximation is a special case of a representation using a combination of Chebyshev and Legendre polynomials. Combining the charged tube and tube in a uniform field allows the solution of voltage driven tubular antennas. Comparisons are made with numerical solutions using piecewise sinusoidal representations of the current. The results are also generalized to the dynamic case and up to and beyond the first resonance. Simple corrections for finite gap and delta gap drives to magnetic frill drives are examined using infinite tube integral transform representations. Corrections between magnetic frill drives and coaxial drives are also given. Approximate drive corrections using conformal mapping and an effective radius are also discussed. Finally, this efficient current representation is applied to the magnetic problem involving simple tubular solenoids.

97 MATHEMATICS AND COMPUTING↗