SEARCH · Search NASA

Results for “Collocation scheme”

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.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3

All sources NASA NTRS DOE OSTI

NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠− 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK DG, collocation, and CG methods.

Numerical Methods for Ordinary Differential Equati↗

NASA NTRS2003

Uncertainty Propagation for Turbulent, Compressible Flow in a Quasi-1D Nozzle Using Stochastic Methods

This paper describes a fully spectral, Polynomial Chaos method for the propagation of uncertainty in numerical simulations of compressible, turbulent flow, as well as a novel stochastic collocation algorithm for the same application. The stochastic collocation method is key to the efficient use of stochastic methods on problems with complex nonlinearities, such as those associated with the turbulence model equations in compressible flow and for CFD schemes requiring solution of a Riemann problem. Both methods are applied to compressible flow in a quasi-one-dimensional nozzle. The stochastic collocation method is roughly an order of magnitude faster than the fully Galerkin Polynomial Chaos method on the inviscid problem.

Zang, Thomas A.↗

NASA NTRS1980

Stability of nonuniform rotor blades in hover using a mixed formulation

A mixed formulation for calculating static equilibrium and stability eigenvalues of nonuniform rotor blades in hover is presented. The static equilibrium equations are nonlinear and are solved by an accurate and efficient collocation method. The linearized perturbation equations are solved by a one step, second order integration scheme. The numerical results correlate very well with published results from a nearly identical stability analysis based on a displacement formulation. Slight differences in the results are traced to terms in the equations that relate moments to derivatives of rotations. With the present ordering scheme, in which terms of the order of squares of rotations are neglected with respect to unity, it is not possible to achieve completely equivalent models based on mixed and displacement formulations. The one step methods reveal that a second order Taylor expansion is necessary to achieve good convergence for nonuniform rotating blades. Numerical results for a hypothetical nonuniform blade, including the nonlinear static equilibrium solution, were obtained with no more effort or computer time than that required for a uniform blade.

Stephens, W. B.↗

NASA NTRS1992

Fourier/Chebyshev methods for the incompressible Navier-Stokes equations in finite domains

A fully spectral numerical scheme for the incompressible Navier-Stokes equations in domains which are infinite or semi-infinite in one dimension. The domain is not mapped, and standard Fourier or Chebyshev expansions can be used. The handling of the infinite domain does not introduce any significant overhead. The scheme assumes that the vorticity in the flow is essentially concentrated in a finite region, which is represented numerically by standard spectral collocation methods. To accomodate the slow exponential decay of the velocities at infinity, extra expansion functions are introduced, which are handled analytically. A detailed error analysis is presented, and two applications to Direct Numerical Simulation of turbulent flows are discussed in relation with the numerical performance of the scheme.

Corral, Roque↗

NASA NTRS1989

Direct simulation of compressible turbulence

Several direct simulations of 3-D homogeneous, compressible turbulence are presented with emphasis on the differences with incompressible turbulent simulations. A fully spectral collocation algorithm, periodic in all directions coupled with a 3rd order Runge-Kutta time discretization scheme is sufficient to produce well-resolved flows at Taylor Reynolds numbers below 40 on grids of 128x128x128. A Helmholtz decomposition of velocity is useful to differentiate between the purely compressible effects and those effects solely due to vorticity production. In the context of homogeneous flows, this decomposition in unique. Time-dependent energy and dissipation spectra of the compressible and solenoidal velocity components indicate the presence of localized small scale structures. These structures are strongly a function of the initial conditions. Researchers concentrate on a regime characterized by very small fluctuating Mach numbers Ma (on the order of 0.03) and density and temperature fluctuations much greater than sq Ma. This leads to a state in which more than 70 percent of the kinetic energy is contained in the so-called compressible component of the velocity. Furthermore, these conditions lead to the formation of curved weak shocks (or shocklets) which travel at approximately the sound speed across the physical domain. Various terms in the vorticity and divergence of velocity production equations are plotted versus time to gain some understanding of how small scales are actually formed. Possible links with Burger turbulence are examined. To visualize better the dynamics of the flow, new graphic visualization techniques have been developed. The 3-D structure of the shocks are visualized with the help of volume rendering algorithms developed in-house. A combination of stereographic projection and animation greatly increase the number of visual cues necessary to properly interpret the complex flow.

Zang, T. A.↗

NASA NTRS2001

A Multiyear Dataset of SSM/I-Derived Global Ocean Surface Turbulent Fluxes

The surface turbulent fluxes of momentum, latent heat, and sensible heat over global oceans are essential to weather, climate and ocean problems. Evaporation is a key component of the hydrological cycle and the surface heat budget, while the wind stress is the major forcing for driving the oceanic circulation. The global air-sea fluxes of momentum, latent and sensible heat, radiation, and freshwater (precipitation-evaporation) are the forcing for driving oceanic circulation and, hence, are essential for understanding the general circulation of global oceans. The global air-sea fluxes are required for driving ocean models and validating coupled ocean-atmosphere global models. We have produced a 7.5-year (July 1987-December 1994) dataset of daily surface turbulent fluxes over the global oceans from the Special Sensor microwave/Imager (SSM/I) data. Daily turbulent fluxes were derived from daily data of SSM/I surface winds and specific humidity, National Centers for Environmental Prediction (NCEP) sea surface temperatures, and European Centre for Medium-Range Weather Forecasts (ECMWF) air-sea temperature differences, using a stability-dependent bulk scheme. The retrieved instantaneous surface air humidity (with a 25-km resolution) validated well with that of the collocated radiosonde observations over the global oceans. Furthermore, the retrieved daily wind stresses and latent heat fluxes were found to agree well with that of the in situ measurements (IMET buoy, RV Moana Wave, and RV Wecoma) in the western Pacific warm pool during the TOGA COARE intensive observing period (November 1992-February 1993). The global distributions of 1988-94 seasonal-mean turbulent fluxes will be presented. In addition, the global distributions of 1990-93 annual-means turbulent fluxes and input variables will be compared with those of UWM/COADS covering the same period. The latter is based on the COADS (comprehensive ocean-atmosphere data set) and is recognized to be one of the best climatological analyses of fluxes derived from ship observations.

Chou, Shu-Hsien↗

NASA NTRS2001

Distributed Relaxation for Conservative Discretizations

A multigrid method is defined as having textbook multigrid efficiency (TME) if the solutions to the governing system of equations are attained in a computational work that is a small (less than 10) multiple of the operation count in one target-grid residual evaluation. The way to achieve this efficiency is the distributed relaxation approach. TME solvers employing distributed relaxation have already been demonstrated for nonconservative formulations of high-Reynolds-number viscous incompressible and subsonic compressible flow regimes. The purpose of this paper is to provide foundations for applications of distributed relaxation to conservative discretizations. A direct correspondence between the primitive variable interpolations for calculating fluxes in conservative finite-volume discretizations and stencils of the discretized derivatives in the nonconservative formulation has been established. Based on this correspondence, one can arrive at a conservative discretization which is very efficiently solved with a nonconservative relaxation scheme and this is demonstrated for conservative discretization of the quasi one-dimensional Euler equations. Formulations for both staggered and collocated grid arrangements are considered and extensions of the general procedure to multiple dimensions are discussed.

Diskin, Boris↗

NASA NTRS2001

A 7.5-Year Dataset of SSM/I-Derived Surface Turbulent Fluxes Over Global Oceans

The surface turbulent fluxes of momentum, latent heat, and sensible heat over global oceans are essential to weather, climate and ocean problems. Wind stress is the major forcing for driving the oceanic circulation, while Evaporation is a key component of hydrological cycle and surface heat budget. We have produced a 7.5-year (July 1987-December 1994) dataset of daily, individual monthly-mean and climatological (1988-94) monthly-mean surface turbulent fluxes over the global oceans from measurements of the Special Sensor Microwave/Imager (SSM/I) on board the US Defense Meteorological Satellite Program F8, F10, and F11 satellites. It has a spatial resolution of 2.0x2.5 latitude-longitude. Daily turbulent fluxes are derived from daily data of SSM/I surface winds and specific humidity, National Centers for Environmental Prediction (NCEP) sea surface temperatures, and European Centre for Medium-Range Weather Forecasts (ECMWF) air-sea temperature differences, using a stability-dependent bulk scheme. The retrieved instantaneous surface air humidity (with a 25-km resolution) IS found to be generally accurate as compared to the collocated radiosonde observations over global oceans. The surface wind speed and specific humidity (latent heat flux) derived from the F10 SSM/I are found to be -encrally smaller (larger) than those retrieved from the F11 SSM/I. The F11 SSM/I appears to have slightly better retrieval accuracy for surface wind speed and humidity as compared to the F10 SSM/I. This difference may be due to the orbital drift of the F10 satellite. The daily wind stresses and latent heat fluxes retrieved from F10 and F11 SSM/Is show useful accuracy as verified against the research quality in si -neasurerrients (IMET buoy, RV Moana Wave, and RV Wecoma) in the western Pacific warm pool during the TOGA COARE Intensive observing period (November 1992-February 1993). The 1988-94 seasonal-mean turbulent fluxes and input variables derived from FS and F11 SSM/Is show reasonable patterns related to seasonal variations of atmospheric general circulation. This dataset of SSM/I-derived turbulent fluxes is useful for climate studies, forcing of ocean models, and validation of coupled ocean-atmosphere global models and can be accessed through the NASA/GSFC Distributed Active Archive Center.

Chou, Shu-Hsien↗

NASA NTRSDate not supplied

Variable Lidar Ratios for Marine and Dusty Marine Using MODIS AOD Constrained Retrievals and GOCART Model Simulations and their Impact on Level 2 CALIPSO Data Products in Version 5

The Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) onboard CALIPSO provided global measurements of attenuated backscatter profiles of various tropospheric aerosol species from June 2006 through June 2023. Extinction profiles are retrieved from these backscatter measurements by assuming a value for the extinction-to-backscatter or lidar ratio (LR). Up until version V4.51 of the data products, only a constant global value for each of the species has been used, spatially and temporally. However, it has been well known from various ground-based measurements that LRs of each aerosol species can potentially vary spatially as well seasonally. For the forthcoming Version 5 of the CALIPSO products, the CALIPSO team will implement variable lidar ratios at 532 nm for the species classified as “marine” and “dusty marine”. Here we present the essential elements of this scheme, which involves estimating lidar ratios by constraining Fernald solutions for the CALIOP particulate backscatter profiles using aerosol optical depths (AOD) from collocated MODIS retrievals. To account for the lack of measurements in various regions and seasons (e.g., Arctic winters), we leverage collocated model sea-salt volume fractions (SSVF) simulated by the Goddard Chemistry Aerosol Radiation and Transport (GOCART) model and use an empirical relationship between the SSVF and LR to build hybrid climatological maps for the entire globe, i.e., using the modelled values when reliable values of the constrained LRs are not available. Initial evaluation of the resulting extinction profiles and corresponding AODs, both globally and in specific regions are presented using a limited amount of data. The primary improvement in AOD occurs in the coastal areas, like the Arabian Sea and Bay of Bengal. The estimated marine LR values in these regions are significantly higher than the constant value of 23 sr used in previous data releases, likely due to mixing with offshore pollution which results in higher AODs. Initial validation results using the CALIPSO Ocean Derived Column Optical Depth (ODCOD) retrievals and ground-based AERONET AOD measurements are presented.

CALIPSO↗

NASA NTRS1988

On the coupling of hyperbolic and parabolic systems: Analytical and numerical approach

The coupling of hyperbolic and parabolic systems is discussed in a domain Omega divided into two distinct subdomains omega(+) and omega(-). The main concern is to find the proper interface conditions to be fulfilled at the surface separating the two domains. Next, they are used in the numerical approximation of the problem. The justification of the interface conditions is based on a singular perturbation analysis, i.e., the hyperbolic system is rendered parabolic by adding a small artifical viscosity. As this goes to zero, the coupled parabolic-parabolic problem degenerates into the original one, yielding some conditions at the interface. These are taken as interface conditions for the hyperbolic-parabolic problem. Actually, two alternative sets of interface conditions are discussed according to whether the regularization procedure is variational or nonvariational. It is shown how these conditions can be used in the frame of a numerical approximation to the given problem. Furthermore, a method of resolution is discussed which alternates the resolution of the hyperbolic problem within omega(-) and of the parabolic one within omega(+). The spectral collocation method is proposed, as an example of space discretization (different methods could be used as well); both explicit and implicit time-advancing schemes are considered. The present study is a preliminary step toward the analysis of the coupling between Euler and Navier-Stokes equations for compressible flows.

Gastaldi, Fabio↗

NASA NTRS2011

Collocation and Galerkin Time-Stepping Methods

We study the numerical solutions of ordinary differential equations by one-step methods where the solution at tn is known and that at t(sub n+1) is to be calculated. The approaches employed are collocation, continuous Galerkin (CG) and discontinuous Galerkin (DG). Relations among these three approaches are established. A quadrature formula using s evaluation points is employed for the Galerkin formulations. We show that with such a quadrature, the CG method is identical to the collocation method using quadrature points as collocation points. Furthermore, if the quadrature formula is the right Radau one (including t(sub n+1)), then the DG and CG methods also become identical, and they reduce to the Radau IIA collocation method. In addition, we present a generalization of DG that yields a method identical to CG and collocation with arbitrary collocation points. Thus, the collocation, CG, and generalized DG methods are equivalent, and the latter two methods can be formulated using the differential instead of integral equation. Finally, all schemes discussed can be cast as s-stage implicit Runge-Kutta methods.

Huynh, H. T.↗

NASA NTRSDate not supplied

On Improving the Dust Lidar Ratios for Version 5 of CALIOP by Using Measurements with Source-tagged Model Results

Vertically resolved retrievals of aerosol backscatter and extinction have been available since 2006 from global measurements acquired by the space borne lidar CALIOP onboard the CALIPSO satellite. These retrievals depend crucially upon the ratio of extinction to backscatter (i.e., lidar ratio) which typically has to be assumed for the various aerosol species for an elastic backscatter lidar like CALIOP. For all CALIPSO data releases up until now (version 4.2), these lidar ratios have been taken as constant both spatially and temporally. However, it is well established that the aerosol lidar ratios can vary significantly regionally, seasonally, and also as a function of age and relative humidity. In particular, the dust lidar ratios show significant variation depending upon their source regions, which is related to the mineral composition of dust in that region. In this paper, we shall present a scheme to obtain the regionally and seasonally dependent lidar ratios of dust. In this scheme, which may be implemented in a forthcoming version of CALIOP aerosol extinction retrievals, we first derive lidar ratios of aerosol layers globally, classified as dust by CALIOP, by using the backscatter measurements constrained by collocated MODIS aerosol optical depths. Since this dust will frequently be a mixture of dust from various source regions, we obtain the corresponding regional contributions from the dust simulations using the GEOS model tagged to nine source regions. Finally, combining information from both measurements and models, we obtain the lidar ratios of dust corresponding to the nine source regions. The lidar ratio of the mixed dust anywhere can now be obtained from the tagged model fractions and the corresponding regional lidar ratio. We shall show the application of this scheme for various regions and seasons.

CALIPSO↗

NASA NTRS1999

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

Kouatchou, Jules↗

NASA NTRS1984

Contact stresses in pin-loaded orthotropic plates

The effects of pin elasticity, friction, and clearance on the stresses near the hole in a pin-loaded orthotropic plate are described. The problem is modeled as a contact elasticity problem using complex variable theory, the pin and the plate being two elastic bodies interacting through contact. This modeling is in contrast to previous works which assumed that the pin is rigid or that it exerts a known cosinusoidal radial traction on the hole boundary. Neither of these approaches explicitly involves a pin. A collocation procedure and iteration were used to obtain numerical results for a variety of plate and pin elastic properties and various levels of friction and clearance. Collocation was used to enforce the boundary and iteration was used to find the contact and no-slip regions on the boundary. Details of the numerical scheme are discussed.

Hyer, M. W.↗

NASA NTRS1985

Contact stresses in pin-loaded orthotropic plates

Hyer, M. W.↗

NASA NTRS2011

Fast Simulators for Satellite Cloud Optical Centroid Pressure Retrievals, 1. Evaluation of OMI Cloud Retrievals

We have developed a relatively simple scheme for simulating retrieved cloud optical centroid pressures (OCP) from satellite solar backscatter observations. We have compared simulator results with those from more detailed retrieval simulators that more fully account for the complex radiative transfer in a cloudy atmosphere. We used this fast simulator to conduct a comprehensive evaluation of cloud OCPs from the two OMI algorithms using collocated data from CloudSat and Aqua MODIS, a unique situation afforded by the A-train formation of satellites. We find that both OMI algorithms perform reasonably well and that the two algorithms agree better with each other than either does with the collocated CloudSat data. This indicates that patchy snow/ice, cloud 3D, and aerosol effects not simulated with the CloudSat data are affecting both algorithms similarly. We note that the collocation with CloudSat occurs mainly on the East side of OMI's swath. Therefore, we are not able to address cross-track biases in OMI cloud OCP retrievals. Our fast simulator may also be used to simulate cloud OCP from output generated by general circulation models (GCM) with appropriate account of cloud overlap. We have implemented such a scheme and plan to compare OMI data with GCM output in the near future.

Joiner, J.↗

NASA NTRS1976

Higher-order numerical methods derived from three-point polynomial interpolation

Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and Hermitian finite-difference discretization. The equations generally apply for both uniform and variable meshes. Hybrid schemes resulting from different polynomial approximations for first and second derivatives lead to the nonuniform mesh extension of the so-called compact or Pade difference techniques. A variety of fourth-order methods are described and this concept is extended to sixth-order. Solutions with these procedures are presented for the similar and non-similar boundary layer equations with and without mass transfer, the Burgers equation, and the incompressible viscous flow in a driven cavity. Finally, the interpolation procedure is used to derive higher-order temporal integration schemes and results are shown for the diffusion equation.

Rubin, S. G.↗

NASA NTRS2013

High-Order Entropy Stable Formulations for Computational Fluid Dynamics

A systematic approach is presented for developing entropy stable (SS) formulations of any order for the Navier-Stokes equations. These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality. They are valid for smooth as well as discontinuous flows provided sufficient dissipation is added at shocks and discontinuities. Entropy stable formulations exist for all diagonal norm, summation-by-parts (SBP) operators, including all centered finite-difference operators, Legendre collocation finite-element operators, and certain finite-volume operators. Examples are presented using various entropy stable formulations that demonstrate the current state-of-the-art of these schemes.

Carpenter, Mark H.↗