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NASA NTRS1983

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequence is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iteratively. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Sidi, A.↗

NASA NTRS1985

On the convergence of an iterative formulation of the electromagnetic scattering from an infinite grating of thin wires

Contraction theory is applied to an iterative formulation of electromagnetic scattering from periodic structures and a computational method for insuring convergence is developed. A short history of spectral (or k-space) formulation is presented with an emphasis on application to periodic surfaces. The mathematical background for formulating an iterative equation is covered using straightforward single variable examples including an extension to vector spaces. To insure a convergent solution of the iterative equation, a process called the contraction corrector method is developed. Convergence properties of previously presented iterative solutions to one-dimensional problems are examined utilizing contraction theory and the general conditions for achieving a convergent solution are explored. The contraction corrector method is then applied to several scattering problems including an infinite grating of thin wires with the solution data compared to previous works.

Brand, J. C.↗

NASA NTRS1986

Acceleration of convergence of vector sequences

A general approach to the construction of convergence acceleration methods for vector sequence is proposed. Using this approach, one can generate some known methods, such as the minimal polynomial extrapolation, the reduced rank extrapolation, and the topological epsilon algorithm, and also some new ones. Some of the new methods are easier to implement than the known methods and are observed to have similar numerical properties. The convergence analysis of these new methods is carried out, and it is shown that they are especially suitable for accelerating the convergence of vector sequences that are obtained when one solves linear systems of equations iterative. A stability analysis is also given, and numerical examples are provided. The convergence and stability properties of the topological epsilon algorithm are likewise given.

Sidi, A.↗

NASA NTRS1987

On the relationship between scatterometer-derived convergences and atmospheric moisture

Fields of divergence calculated from the Seasat-A Satellite Scatterometer winds and fields of integrated water vapor and rainrate from the Scanning Multichannel Microwave Radiometer on Seasat are constructed for three different midlatitude cyclones. These storms include an explosively deepening cyclone that occurred in the North Atlantic (also known as the Queen Elizabeth II cyclone), a storm that occurred in the North Pacific, and a Southern Ocean storm. In all three cases, the regions of convergence and atmospheric water (vapor and rain) are consistent with each other and help to define features of each storm. The vertical distribution of moisture is inferred for one case using both the convergence pattern and the integrated water vapor field. In another, interpretation of the convergence field in a data gap region is aided by the water vapor field. In all three cases, surface low pressure centers, fronts, and even frontal waves are clearly evident as areas of convergence, and increased water vapor and rainrate.

Mcmurdie, Lynn A.↗

NASA NTRS1988

Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L(infinity)-stability imply convergence

A convergence theory for semi-discrete approximations to nonlinear systems of conservation laws is developed. It is shown, by a series of scalar counter-examples, that consistency with the conservation law alone does not guarantee convergence. Instead, a notion of consistency which takes into account both the conservation law and its augmenting entropy condition is introduced. In this context it is concluded that consistency and L(infinity)-stability guarantee for a relevant class of admissible entropy functions, that their entropy production rate belongs to a compact subset of H(loc)sup -1 (x,t). One can now use compensated compactness arguments in order to turn this conclusion into a convergence proof. The current state of the art for these arguments includes the scalar and a wide class of 2 x 2 systems of conservation laws. The general framework of the vanishing viscosity method is studied as an effective way to meet the consistency and L(infinity)-stability requirements. How this method is utilized to enforce consistency and stability for scalar conservation laws is shown. In this context we prove, under the appropriate assumptions, the convergence of finite difference approximations (e.g., the high resolution TVD and UNO methods), finite element approximations (e.g., the Streamline-Diffusion methods) and spectral and pseudospectral approximations (e.g., the Spectral Viscosity methods).

Tadmor, Eitan↗

NASA NTRS1988

Convergence acceleration for vector sequences and applications to computational fluid dynamics

Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.

Sidi, Avram↗

NASA NTRS1990

Convergence acceleration for vector sequences and applications to computational fluid dynamics

Sidi, Avram↗

NASA NTRS1991

Surface wind convergence as a short-term predictor of cloud-to-ground lightning at Kennedy Space Center

Cloud-to-ground lightning is a significant forecast problem at the Kennedy Space Center (KSC) in Florida. In this study, cloud-to-ground lightning is related in time and space to surface convergence for 244 days during the convective seasons of 1985 and 1986 over a 790 sq-km network at KSC. The method uses surface convergence, particularly the average over the area, to identify the potential for new, local thunderstorm growth, and it can be used to specify the likely time and location of lightning during the life cycle of the convection. A threshold of 75 x 10 to the -6th/s change in divergence is the main criterion used to define a convergence event, and a set of flashes less than 30 min apart defines a lightning event. Time intervals are found from the study to be approximately 1 h from beginning convergence to first flash, and another hour to the end of lightning. The influences of low-level winds and midlevel moisture in determining the location and intensity of convection are discussed. This is the first known dynamically-based forecast method for lightning prediction. The technique, currently in use at KSC, has been shown to be a systematic, quantitative tool for predicting lightning onset in situations where conventional analysis tools such as radar and satellite are limited.

Watson, Andrew I.↗

NASA NTRS1991

Mantle downwelling and crustal convergence - A model for Ishtar Terra, Venus

Models of viscous crustal flow driven by gradients in topography are presented in order to explore quantitatively the implications of the hypothesis that Ishtar is a crustal convergence zone overlying a downwelling mantle. Assuming a free-slip surface boundary condition, it is found that, if the crustal convergence hypothesis is correct, then the crustal thickness in the plains surrounding Ishtar can be no more than about 25 km thick. If the geothermal gradient is larger or the rheology is weaker, the crust must be even thinner for net crustal convergence to be possible. This upper bound is in good agreement with the several independent estimates of crustal thickness of 15-30 km in the plains of Venus based on modeling of the spacing of tectonic features and of impact crater relaxation. Although Ishtar is treated as a crustal convergence zone, this crustal flow model shows that under some circumstances, near-surface material may actually flow away from Ishtar, providing a possible explanation for the grabenlike structures in Fortuna Tessera.

Kiefer, Walter S.↗

NASA NTRS1993

The effect of grid aspect ratio on convergence

Vector stability theory and computational experiments are used to examine the effects of high aspect ratio grids on algorithm convergence. Emphasis is placed on identifying the degree of convergence deterioration that ADI schemes experience as grid aspect ratio increases and on deriving methods for circumventing this problem. Approximately factored implicit schemes are found to experience convergence deterioration because of nonoptimum local time stepping procedures and the increased need for viscous preconditioning. An enhanced algorithm is devised on the basis of improved selection of the local time step, appropriate definition of viscous preconditioning matrix, and proper implementation of boundary conditions. The novel algorithm provides uniformly efficient convergence at all aspect ratios for both Euler and Navier-Stokes computations for a variety of test problems.

Buelow, P. E. O.↗

NASA NTRS1994

Preferred latitudes of the intertropical convergence zone

The latitude preference of the intertropical convergence zone (ITCZ) is examined on the basis of observations, theory, and a modeling analysis. Observations show that convection is enhanced at latitudes of about 4 deg to 10 deg relative to the equator, even in regions where the sea surface temperature (SST) is maximum on the equator. Both linear shallow-water theory and a moist primitive equation model suggest a new explanation for the off-equatorial latitude preference of the ITCZ that requires neither the existence of zonally propagating disturbances nor an off-equatorial maximum in SST. The shallow-water theory indicates that a finite-width, zonally oriented, midtropospheric heat source (i.e., an ITCZ) produces the greatest local low-level convergence when placed a finite distance away from the equator. This result suggests that an ITCZ is most likely to be supported via low-level convergence of moist energy when located at these "preferred" latitudes away from the equator. For a plausible range of heating widths and damping parameters, the theoretically predicted latitude is approximately equal to the observed position (s) of the ITCZ (s). Analysis with an axially symmetric, moist, primitive equation model indicates that when the latent heating field is allowed to be determined internally, a positive feedback develops between the midtropospheric latent heating and the low-level convergence, with the effect of enhancing the organization of convection at latitudes of about 4 deg to 12 deg. Numerical experiments show that (1) two peaks in convective precipitation develop straddling the equator when the SST maximum is located on the equator; (2) steady ITCZ-like structures form only when the SST maximum is located away from the equator; and (3) peaks in convection can develop away from the maximum in SST, with a particular preference for latitudes of about 4 deg to 12 deg, even in the ('cold') hemisphere without the SST maximum. The relationship between this mechanism and earlier theories is discussed, as are implications for the coupled ocean-atmosphere system and the roles played by midlevel latent heating and SST gradients in forcing the low-level atmospheric circulation in the tropics.

Waliser, Duane E.↗

NASA NTRS2001

Idealized Simulations of the Effects of Amazon Convection and Baroclinic Waves on the South Atlantic Convergence Zone

The South Atlantic Convergence Zone (SACZ) is a NW-SE oriented, stationary region of enhanced convergence and convection that extends southeastward from the ITCZ convection anchored over the Amazon region. On daily satellite images each SACZ episode is seen as a progression of one or several midlatitude cold fronts that intrude into the subtropics and tropics, becoming stationary over southeastern Brazil for a few days. Previous studies have shown that while Amazon convection plays a fundamental role in the formation of the SACZ, Atlantic sea surface temperatures and the Andes Mountains play a relatively minor role in the strength and location of the SACZ. The role of interactions between Amazon convection and midlatitude baroclinic waves in establishing the origin, position, and maintenance of the SACZ is studied here using idealized dry, multilayer global model simulations that do not include the effects of topography. The model simulations produce SACZ-like regions of low-level convergence in the presence of Amazon convection embedded in a mean-flow that contains propagating baroclinic waves. The results of these simulations indicate that Amazon convection plays two fundamental roles in the formation and location of the SACZ. First, it produces a NW-SE oriented region of low-level convergence to the SE of Amazon convection. Second, it produces a storm-track region and accompanying stronger midlatitude baroclinic waves in the region of the SACZ. It is suggested that in the presence of moist effects, the 'seedling' SACZ regions produced in these simulations can be enhanced to produce the observed SACZ.

Ferreira, Rosana Nieto↗

NASA NTRS1995

Binocular Convergence and Errors in Judged Distance While Using Head-mounted See-through Displays

Previous observations have shown that optical superposition of a physical backdrops at the judged depth of a stereoscopic virtual image moves the judged depth closer to the observer. This effect was more pronounced for slowly moving physical backdrops and was not enhanced when the virtual image was rendered as a flat shaded solid object rather than a open wire-frame. Since this change in rendering making the virtual image more completely occlude the backdrop did not effect its judged depth and since the motion of the backdrop which would have attracted visual attention and binocular convergence did Increase its perceptual displacement, it was concluded that the change In Judged depth was not due to the perceived occlusion. Rather it was concluded to be due to an increase in binocular convergence. An experimental test of this hypothesis using a unobtrusive nonius technique to detect absolute and relative convergence has confirmed the presence of convergence correlated with the magnitude of the change in judged position of this virtual Image. The practical implications of this cause are demonstrated by a second study using monocular, biocular and stereoscopic viewing conditions and the consequences for the design of head-mounted see-through displays for near work are discussed.

Ellis, Stephen R.↗

NASA NTRS2002

On the Numerical Convergence to Steady State of Hypersonic Flows Over Bodies with Concavities

Two recent numerical studies of hypersonic flows over bodies with concavities revealed problems with convergence to a steady state with an oft-used application of local-time-stepping. Both simulated flows showed a time-like, periodic shedding of vortices in a subsonic domain bounded by supersonic external flow although the simulations, using local-time-stepping, were not time accurate. Simple modifications to the numerical algorithm were implemented to enable implicit, first-order accurate in time simulations. Subsequent time-accurate simulations of the two test problems converged to a steady state. The baseline algorithm and modifications for temporal accuracy are described. The requirement for sub-iterations to achieve convergence is demonstrated. Failure to achieve convergence without time accuracy is conjectured to arise from temporal errors being continuously refocused into a subsonic domain.

Gnoffo, Peter A.↗

NASA NTRS2003

Experimental and Computational Investigation of Multiple Injection Ports in a Convergent-Divergent Nozzle for Fluidic Thrust Vectoring

A computational and experimental study was conducted to investigate the effects of multiple injection ports in a two-dimensional, convergent-divergent nozzle, for fluidic thrust vectoring. The concept of multiple injection ports was conceived to enhance the thrust vectoring capability of a convergent-divergent nozzle over that of a single injection port without increasing the secondary mass flow rate requirements. The experimental study was conducted at static conditions in the Jet Exit Test Facility of the 16-Foot Transonic Tunnel Complex at NASA Langley Research Center. Internal nozzle performance was obtained at nozzle pressure ratios up to 10 with secondary nozzle pressure ratios up to 1 for five configurations. The computational study was conducted using the Reynolds Averaged Navier-Stokes computational fluid dynamics code PAB3D with two-equation turbulence closure and linear Reynolds stress modeling. Internal nozzle performance was predicted for nozzle pressure ratios up to 10 with a secondary nozzle pressure ratio of 0.7 for two configurations. Results from the experimental study indicate a benefit to multiple injection ports in a convergent-divergent nozzle. In general, increasing the number of injection ports from one to two increased the pitch thrust vectoring capability without any thrust performance penalties at nozzle pressure ratios less than 4 with high secondary pressure ratios. Results from the computational study are in excellent agreement with experimental results and validates PAB3D as a tool for predicting internal nozzle performance of a two dimensional, convergent-divergent nozzle with multiple injection ports.

Waithe, Kenrick A.↗

NASA NTRS2003

Experimental and Computational Investigation of Multiple Injection Ports in a Convergent-Divergent Nozzle for Fluidic Thrust Vectoring

Waithe, Kenrick A.↗

NASA NTRS2002

Vestibular convergence patterns in vestibular nuclei neurons of alert primates

Sensory signal convergence is a fundamental and important aspect of brain function. Such convergence may often involve complex multidimensional interactions as those proposed for the processing of otolith and semicircular canal (SCC) information for the detection of translational head movements and the effective discrimination from physically congruent gravity signals. In the present study, we have examined the responses of primate rostral vestibular nuclei (VN) neurons that do not exhibit any eye movement-related activity using 0.5-Hz translational and three-dimensional (3D) rotational motion. Three distinct neural populations were identified. Approximately one-fourth of the cells exclusively encoded rotational movements (canal-only neurons) and were unresponsive to translation. The canal-only central neurons encoded head rotation in SCC coordinates, exhibited little orthogonal canal convergence, and were characterized with significantly higher sensitivities to rotation as compared to primary SCC afferents. Another fourth of the neurons modulated their firing rates during translation (otolith-only cells). During rotations, these neurons only responded when the axis of rotation was earth-horizontal and the head was changing orientation relative to gravity. The remaining one-half of VN neurons were sensitive to both rotations and translations (otolith + canal neurons). Unlike primary otolith afferents, however, central neurons often exhibited significant spatiotemporal (noncosine) tuning properties and a wide variety of response dynamics to translation. To characterize the pattern of SCC inputs to otolith + canal neurons, their rotational maximum sensitivity vectors were computed using exclusively responses during earth-vertical axis rotations (EVA). Maximum sensitivity vectors were distributed throughout the 3D space, suggesting strong convergence from multiple SCCs. These neurons were also tested with earth-horizontal axis rotations (EHA), which would activate both vertical canals and otolith organs. However, the recorded responses could not be predicted from a linear combination of EVA rotational and translational responses. In contrast, one-third of the neurons responded similarly during EVA and EHA rotations, although a significant response modulation was present during translation. Thus this subpopulation of otolith + canal cells, which included neurons with either high- or low-pass dynamics to translation, appear to selectively ignore the component of otolith-selective activation that is due to changes in the orientation of the head relative to gravity. Thus contrary to primary otolith afferents and otolith-only central neurons that respond equivalently to tilts relative to gravity and translational movements, approximately one-third of the otolith + canal cells seem to encode a true estimate of the translational component of the imposed passive head and body movement.

Non-NASA Center↗

NASA NTRS2009

Super-convergence of Discontinuous Galerkin Method Applied to the Navier-Stokes Equations

The practical benefits of the hyper-accuracy properties of the discontinuous Galerkin method are examined. In particular, we demonstrate that some flow attributes exhibit super-convergence even in the absence of any post-processing technique. Theoretical analysis suggest that flow features that are dominated by global propagation speeds and decay or growth rates should be super-convergent. Several discrete forms of the discontinuous Galerkin method are applied to the simulation of unsteady viscous flow over a two-dimensional cylinder. Convergence of the period of the naturally occurring oscillation is examined and shown to converge at 2p+1, where p is the polynomial degree of the discontinuous Galerkin basis. Comparisons are made between the different discretizations and with theoretical analysis.

Atkins, Harold L.↗