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Error in approximation less than computer truncation error. Report discusses scheme for reduction of computational drift in estimation of attitude of spacecraft from strapdown-gyroscope measurements.
The computational fluid dynamics (CFD) prediction workshops sponsored by the AIAA have created invaluable opportunities in which to discuss the predictive capabilities of CFD in areas in which it has struggled, e.g., cruise drag, high-lift, and sonic boom pre diction. While there are many factors that contribute to disagreement between simulated and experimental results, such as modeling or discretization error, quantifying the errors contained in a simulation is important for those who make decisions based on the computational results. The linearized error transport equations (ETE) combined with a truncation error estimate is a method to quantify one source of errors. The ETE are implemented with a complex-step method to provide an exact linearization with minimal source code modifications to CFD and multidisciplinary analysis methods. The equivalency of adjoint and linearized ETE functional error correction is demonstrated. Uniformly refined grids from a series of AIAA prediction workshops demonstrate the utility of ETE for multidisciplinary analysis with a connection between estimated discretization error and (resolved or under-resolved) flow features.
Existing frequency domain modal frequency and damping analysis methods are discussed. The effects of truncation in the Laplace and Fourier transform data analysis methods are described. Methods for eliminating truncation errors from measured damping are presented. Implications of truncation effects in fast Fourier transform analysis are discussed. Limited comparison with test data is presented.
Improved knowledge of the Earth's gravity field was obtained from new and improved satellite measurements such as satellite to satellite tracking and gradiometry. This improvement was examined by estimating the accuracy of the determination of mean anomalies and mean undulations in various size blocks based on an assumed mission. In this report the accuracy is considered through a commission error due to measurement noise propagation and a truncation error due to unobservable higher degree terms in the geopotential. To do this the spectrum of the measurement was related to the spectrum of the disturbing potential of the Earth's gravity field. Equations were derived for a low-low (radial or horizontal separation) mission and a gradiometer mission. For a low-low mission of six month's duration, at an altitude of 160 km, with a data noise of plus or minus 1 micrometers sec for a four second integration time, we would expect to determine 1 deg x 1 deg mean anomalies to an accuracy of plus or minus 2.3 mgals and 1 deg x 1 deg mean geoid undulations to plus or minus 4.3 cm. A very fast Fortran program is available to study various mission configurations and block sizes.
The uncertainty in the geopotential model is currently one of the major error sources in the orbit determination of low-altitude Earth-orbiting spacecraft. The results of an investigation of different geopotential error models and modeling approaches currently used for operational orbit error analysis support at the Goddard Space Flight Center (GSFC) are presented, with emphasis placed on sequential orbit error analysis using a Kalman filtering algorithm. Several geopotential models, known as the Goddard Earth Models (GEMs), were developed and used at GSFC for orbit determination. The errors in the geopotential models arise from the truncation errors that result from the omission of higher order terms (omission errors) and the errors in the spherical harmonic coefficients themselves (commission errors). At GSFC, two error modeling approaches were operationally used to analyze the effects of geopotential uncertainties on the accuracy of spacecraft orbit determination - the lumped error modeling and uncorrelated error modeling. The lumped error modeling approach computes the orbit determination errors on the basis of either the calibrated standard deviations of a geopotential model's coefficients or the weighted difference between two independently derived geopotential models. The uncorrelated error modeling approach treats the errors in the individual spherical harmonic components as uncorrelated error sources and computes the aggregate effect using a combination of individual coefficient effects. This study assesses the reasonableness of the two error modeling approaches in terms of global error distribution characteristics and orbit error analysis results. Specifically, this study presents the global distribution of geopotential acceleration errors for several gravity error models and assesses the orbit determination errors resulting from these error models for three types of spacecraft - the Gamma Ray Observatory, the Ocean Topography Experiment, and the Cosmic Background Explorer.
The laser velocimeter (LV) technique has so far been successfully applied to many flow situations to obtain mean velocity and turbulence level measurements. The paper deals with a unique LV system employing a digital frequency counter for measurements of turbulence spectra within the objectives of achieving higher accuracies and a wider range of velocities than were possible by previous analog systems. Systematic analysis of the effects of LV system errors in the accuracy of the turbulence power spectral measurements is performed and shown to agree reasonably with the experimental LV spectra results. Sufficient understanding of the accuracy and limitations of the digital frequency counter LV spectral measurement system is obtained. Among the major system errors encountered in a typical digital frequency counter LV processor, the truncation error is regarded as the most serious source of error. The quantizing step size defined as the ratio of Doppler frequency to processor clock frequency is an important parameter in determining the dynamic range of a turbulence power spectrum.
The Galerkin-Weighted Residuals formulation is employed to derive an implicit finite element solution algorithm for the nonlinear parabolic partial differential equation system governing turbulent boundary layer flow. Solution accuracy and convergence with discretization refinement are quantized in several error norms using linear and quadratic basis functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the nonlinear equations characteristic of turbulent boundary layer flow.
The adaptive solution of the Navier-Stokes equations depends upon the successful interaction of three key elements: (1) the ability to flexibly select grid length scales in composite grids, (2) the ability to efficiently control residual error in composite grids, and (3) the ability to define reliable, convenient error norms to guide the grid adjustment and optimize the residual levels relative to the local truncation errors. An initial investigation was conducted to explore how to approach developing these key elements. Conventional error assessment methods were defined and defect and deferred correction methods were surveyed. The one dimensional potential equation was used as a multigrid test bed to investigate how to achieve successful interaction of these three key elements.
Curvilinear coordinate systems were used extensively to solve partial differential equations on arbitrary regions. An analysis of truncation error in the computation of derivatives revealed why numerical results may be erroneous. A more accurate method of computing derivatives is presented.
The cell-vertex schemes due to Ni and Jameson, et al. have been subjected to a theoretical analysis of their truncation error. The analysis confirms the authors' claims for second-order accuracy on smooth grids, but shows that the same accuracy cannot be obtained on arbitrary grids. It is shown that the schemes have a unique generalization to axisymmetric flow that preserves the second-order accuracy.
Assessing solution error continues to be a formidable task when numerically solving practical flow problems. Currently, grid refinement is the primary method used for error assessment. The minimum grid spacing requirements to achieve design order accuracy for a structured-grid scheme are determined for several simple examples using truncation error evaluations on a sequence of meshes. For certain methods and classes of problems, obtaining design order may not be sufficient to guarantee low error. Furthermore, some schemes can require much finer meshes to obtain design order than would be needed to reduce the error to acceptable levels. Results are then presented from realistic problems that further demonstrate the challenges associated with using grid refinement studies to assess solution accuracy.
The Galerkin-weighted residuals formulation is employed to derive an implicit finite element solution algorithm for a generally non-linear initial-boundary value problem. Solution accuracy and convergence with discretization refinement are quantized in several error norms, for the non-linear parabolic partial differential equation system governing laminar boundary layer flow, using linear, quadratic and cubic functions. Richardson extrapolation is used to isolate integration truncation error in all norms, and Newton iteration is employed for all equation solutions performed in double-precision. The mathematical theory supporting accuracy and convergence concepts for linear elliptic equations appears extensible to the non-linear equations characteristic of laminar boundary layer flow.
Optimal control techniques are used to drive the numerical error (truncation, roundoff, commutation) in computing the quaternion vector to zero. The normalization of the quaternion is carried out by appropriate choice of a performance index, which can be optimized. The error equations are derived from Friedland's (1978) theoretical development, and a matrix Riccati equation results for the computation of the gain matrix. Simulation results show that a high precision of the order of 10 to the -12th can be obtained using this technique in meeting the q(T)q=1 constraint. The performance of the estimator in the presence of the feedback control that maintains the normalization, is studied.
Errors in the outer zone contribution to oceanic undulation differences computed from a finite set of potential coefficients based on satellite measurements of gravity anomalies and gravity disturbances are analyzed. Equations are derived for the truncation errors resulting from the lack of high-degree coefficients and the commission errors arising from errors in the available lower-degree coefficients, and it is assumed that the inner zone (spherical cap) is sufficiently covered by surface gravity measurements in conjunction with altimetry or by gravity anomaly data. Numerical computations of error for various observational conditions reveal undulation difference errors ranging from 13 to 15 cm and from 6 to 36 cm in the cases of gravity anomaly and gravity disturbance data, respectively for a cap radius of 10 deg and mean anomalies accurate to 10 mgal, with a reduction of errors in both cases to less than 10 cm as mean anomaly accuracy is increased to 1 mgal. In the absence of a spherical cap, both cases yield error estimates of 68 cm for an accuracy of 1 mgal and between 93 and 160 cm for the lesser accuracy, which can be reduced to about 110 cm by the introduction of a perfect 30-deg reference field.
Presented here is an algorithm for solving the multidimensional unsteady Navier-Stokes equations for compressible flows. It is based on a diagonally-dominant approximate factorization procedure. The factorization error and the timewise linearization error associated with this procedure are reduced by performing Newton-type inner iterations at each time step. The inviscid fluxes are evaluated by the fourth-order central differencing scheme amended with a numerical dissipation directly proportional to the entire dissipative part of the truncation error intrinsic to the third order biased upwind scheme. The important features of the proposed solution are elucidated by the numerical results of the convection of a vortex and the backward-facing step flows.
Errors in computations of the geoid undulation, which determines the position of the geoid with respect to a reference ellipsoid, introduced by various modifications of the Stokes' function in the Stokes integral formula for the undulation are examined in light of currently available harmonic coefficients. The variants considered include the conventional Molodenskii truncation theory, in which integration is limited to a spherical cap, and the variations suggested by Molodenskii (1958) and Meissl (1971). In computations of the undulation from gravity anomalies in a spherical cap and terrestrial and satellite harmonic potential coefficients (GEM 9 and GEM 10B), it is shown that either of the modifications to the conventional Molodenskii theory results in substantial reductions in truncation errors, with the simpler modification of Meissl even providing superior results in certain circumstances.
A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.
The implementation of a technique for locating programming errors in shallow-water codes, establishing the correctness of the code, and assessing the performance of the numerical model under various flow conditions is described. The right-hand side of the differential equations is modified in such a way that the exact solution of the nonlinear initial-value problem is known, so that the truncation errors of the numerical scheme can be studied in detail. The exact solution is prescribed to be any linear combination of Hough harmonics which propagate in time according to their natural frequencies.