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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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NASA NTRS1977

Recurrence relations for computing with modified divided differences

Modified divided differences (MDD) provide a good way of representing a polynomial passing through points with unequally spaced abcissas. Recurrence relations for computing coefficients in either the monomial or Chebyshev basis from the MDD coefficients, and for computing the MDD coefficients for either the differentiated or the integrated polynomial are given. The latter operation is likely to be useful if MDD are used in a method for solving stiff differential equations.

Krogh, F.↗
NASA NTRS1979

Recurrence relations for computing with modified divided differences

Modified divided differences (MDD) provide a good way of representing a polynomial passing through points with unequally spaced abscissas. This note gives recurrence relations for computing coefficients in either the monomial or Chebyshev basis from the MDD coefficients, and for computing the MDD coefficients for either the differentiated or the integrated polynomial. The latter operation is likely to be useful if MDD are used in a method for solving stiff differential equations.

Krogh, F. T.↗
NASA NTRS1974

Changing stepsize in the integration of differential equations using modified divided differences

Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.

Krogh, F. T.↗
NASA NTRS1993

Sensitivity derivatives for advanced CFD algorithm and viscous modelling parameters via automatic differentiation

The computational technique of automatic differentiation (AD) is applied to a three-dimensional thin-layer Navier-Stokes multigrid flow solver to assess the feasibility and computational impact of obtaining exact sensitivity derivatives typical of those needed for sensitivity analyses. Calculations are performed for an ONERA M6 wing in transonic flow with both the Baldwin-Lomax and Johnson-King turbulence models. The wing lift, drag, and pitching moment coefficients are differentiated with respect to two different groups of input parameters. The first group consists of the second- and fourth-order damping coefficients of the computational algorithm, whereas the second group consists of two parameters in the viscous turbulent flow physics modelling. Results obtained via AD are compared, for both accuracy and computational efficiency with the results obtained with divided differences (DD). The AD results are accurate, extremely simple to obtain, and show significant computational advantage over those obtained by DD for some cases.

Green, Lawrence L.↗
NASA NTRS1994

Applications of automatic differentiation in computational fluid dynamics

Automatic differentiation (AD) is a powerful computational method that provides for computing exact sensitivity derivatives (SD) from existing computer programs for multidisciplinary design optimization (MDO) or in sensitivity analysis. A pre-compiler AD tool for FORTRAN programs called ADIFOR has been developed. The ADIFOR tool has been easily and quickly applied by NASA Langley researchers to assess the feasibility and computational impact of AD in MDO with several different FORTRAN programs. These include a state-of-the-art three dimensional multigrid Navier-Stokes flow solver for wings or aircraft configurations in transonic turbulent flow. With ADIFOR the user specifies sets of independent and dependent variables with an existing computer code. ADIFOR then traces the dependency path throughout the code, applies the chain rule to formulate derivative expressions, and generates new code to compute the required SD matrix. The resulting codes have been verified to compute exact non-geometric and geometric SD for a variety of cases. in less time than is required to compute the SD matrix using centered divided differences.

Green, Lawrence L.↗
NASA NTRS1979

Nonlinear wave choked inlets

The quasi-one dimensional flow program was modified in two ways. The Runge-Kutta subroutine was replaced with a subroutine which used a modified divided difference form of the Adams Pece method and the matrix inversion routine was replaced with a pseudo inverse routine. Calculations were run using both the original and modified programs. Comparison of the calculations showed that the original Runge-Kutta routine could not detect singularity near the throat and was integrating across it. The modified version was able to detect the singularity and therefore gave more valid calculations.

Source record↗
NASA NTRS1979

Red and photographic infrared linear combinations for monitoring vegetation

The relationships between various linear combinations of red and photographic infrared radiances and vegetation parameters are investigated. In situ spectrometers are used to measure the relationships between linear combinations of red and IR radiances, their ratios and square roots, and biomass, leaf water content and chlorophyll content of a grass canopy in June, September and October. Regression analysis shows red-IR combinations to be more significant than green-red combinations. The IR/red ratio, the square root of the IR/red ratio, the vegetation index (IR-red difference divided by their sum) and the transformed vegetation index (the square root of the vegetation index + 0.5) are found to be sensitive to the amount of photosynthetically active vegetation. The accumulation of dead vegetation over the year is found to have a linearizing effect on the various vegetation measures.

Tucker, C. J.↗
NASA NTRS1983

The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme

The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = 0 where f(u) is, say, strictly convex double dot f dot a sub asterisk 0 is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to liner problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitive insight into the large-time behavior of the numerical computation.

Tadmor, E.↗
NASA NTRS1984

The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme

The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = O where f(u) is, say, strictly double dot f dot a sub asterisk O is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to linear problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitative insight into the large-time behavior of the numerical computation.

Tadmor, E.↗
NASA NTRS1988

High degree interpolation polynomial in Newton form

Polynomial interpolation is an essential subject in numerical analysis. Dealing with a real interval, it is well known that even if f(x) is an analytic function, interpolating at equally spaced points can diverge. On the other hand, interpolating at the zeroes of the corresponding Chebyshev polynomial will converge. Using the Newton formula, this result of convergence is true only on the theoretical level. It is shown that the algorithm which computes the divided differences is numerically stable only if: (1) the interpolating points are arranged in a different order, and (2) the size of the interval is 4.

Tal-Ezer, Hillel↗
NASA NTRS1996

Development of an Automatic Differentiation Version of the FPX Rotor Code

The ADIFOR2.0 automatic differentiator is applied to the FPX rotor code along with the grid generator GRGN3. The FPX is an eXtended Full-Potential CFD code for rotor calculations. The automatic differentiation version of the code is obtained, which provides both non-geometry and geometry sensitivity derivatives. The sensitivity derivatives via automatic differentiation are presented and compared with divided difference generated derivatives. The study shows that automatic differentiation method gives accurate derivative values in an efficient manner.

Hu, Hong↗
NASA NTRS1998

On the Circulation Manifold for Two Adjacent Lifting Sections

The circulation functional relative to two adjacent lifting sections is studied for two cases. In the first case we consider two adjacent circles. The circulation is computed as a function of the displacement of the secondary circle along the axis joining the two centers and of the angle of attack of the secondary circle, The gradient of such functional is computed by deriving a set of elliptic functions with respect both to their argument and to their Period. In the second case studied, we considered a wing-flap configuration. The circulation is computed by some implicit mappings, whose differentials with respect to the variation of the geometrical configuration in the physical space are found by divided differences. Configurations giving rise to local maxima and minima in the circulation manifold are presented.

Zannetti, Luca↗
NASA NTRS2000

A Very High Order, Adaptable MESA Implementation for Aeroacoustic Computations

Since computational efficiency and wave resolution scale with accuracy, the ideal would be infinitely high accuracy for problems with widely varying wavelength scales. Currently, many of the computational aeroacoustics methods are limited to 4th order accurate Runge-Kutta methods in time which limits their resolution and efficiency. However, a new procedure for implementing the Modified Expansion Solution Approximation (MESA) schemes, based upon Hermitian divided differences, is presented which extends the effective accuracy of the MESA schemes to 57th order in space and time when using 128 bit floating point precision. This new approach has the advantages of reducing round-off error, being easy to program. and is more computationally efficient when compared to previous approaches. Its accuracy is limited only by the floating point hardware. The advantages of this new approach are demonstrated by solving the linearized Euler equations in an open bi-periodic domain. A 500th order MESA scheme can now be created in seconds, making these schemes ideally suited for the next generation of high performance 256-bit (double quadruple) or higher precision computers. This ease of creation makes it possible to adapt the algorithm to the mesh in time instead of its converse: this is ideal for resolving varying wavelength scales which occur in noise generation simulations. And finally, the sources of round-off error which effect the very high order methods are examined and remedies provided that effectively increase the accuracy of the MESA schemes while using current computer technology.

Dydson, Roger W.↗
NASA NTRS2000

An Automated Approach to Very High Order Aeroacoustic Computations in Complex Geometries

Computational aeroacoustics requires efficient, high-resolution simulation tools. And for smooth problems, this is best accomplished with very high order in space and time methods on small stencils. But the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewslci recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that are located near wall boundaries. These procedures are used to automatically develop and implement very high order methods (>15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS2001

Automated Approach to Very High-Order Aeroacoustic Computations

Computational aeroacoustics requires efficient, high-resolution simulation tools. For smooth problems, this is best accomplished with very high-order in space and time methods on small stencils. However, the complexity of highly accurate numerical methods can inhibit their practical application, especially in irregular geometries. This complexity is reduced by using a special form of Hermite divided-difference spatial interpolation on Cartesian grids, and a Cauchy-Kowalewski recursion procedure for time advancement. In addition, a stencil constraint tree reduces the complexity of interpolating grid points that am located near wall boundaries. These procedures are used to develop automatically and to implement very high-order methods (> 15) for solving the linearized Euler equations that can achieve less than one grid point per wavelength resolution away from boundaries by including spatial derivatives of the primitive variables at each grid point. The accuracy of stable surface treatments is currently limited to 11th order for grid aligned boundaries and to 2nd order for irregular boundaries.

Dyson, Rodger W.↗
NASA NTRS2006

Ozone Profiles in the High-latitude Stratosphere and Lower Mesosphere Measured by the Improved Limb Atmospheric Spectrometer (ILAS)-II: Comparison with other Satellite Sensors and Ozonesondes

A solar occultation sensor, the Improved Limb Atmospheric Spectrometer (ILAS)-II, measured 5890 vertical profiles of ozone concentrations in the stratosphere and lower mesosphere and of other species from January to October 2003. The measurement latitude coverage was 54-71degN and 64-88degS, which is similar to the coverage of ILAS (November 1996 to June 1997). One purpose of the ILAS-II measurements was to continue such high-latitude measurements of ozone and its related chemical species in order to help accurately determine their trends. The present paper assesses the quality of ozone data in the version 1.4 retrieval algorithm, through comparisons with results obtained from comprehensive ozonesonde measurements and four satellite-borne solar occultation sensors. In the Northern Hemisphere (NH), the ILAS-II ozone data agree with the other data within +/-10% (in terms of the absolute difference divided by its mean value) at altitudes between 11 and 40 km, with the median coincident ILAS-II profiles being systematically up to 10% higher below 20 km and up to 10% lower between 21 and 40 km after screening possible suspicious retrievals. Above 41 km, the negative bias between the NH ILAS-II ozone data and the other data increases with increasing altitude and reaches 30% at 61-65 km. In the Southern Hemisphere, the ILAS-II ozone data agree with the other data within 10% in the altitude range of 11-60 km, with the median coincident profiles being on average up to 10% higher below 20 km and up to 10% lower above 20 km.

Sugita, T.↗
NASA NTRS2010

Comparison of Sigma-Point and Extended Kalman Filters on a Realistic Orbit Determination Scenario

Sigma-point filters have received a lot of attention in recent years as a better alternative to extended Kalman filters for highly nonlinear problems. In this paper, we compare the performance of the additive divided difference sigma-point filter to the extended Kalman filter when applied to orbit determination of a realistic operational scenario based on the Interstellar Boundary Explorer mission. For the scenario studied, both filters provided equivalent results. The performance of each is discussed in detail.

Gaebler, John↗