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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS2011

Notes on Accuracy of Finite-Volume Discretization Schemes on Irregular Grids

Truncation-error analysis is a reliable tool in predicting convergence rates of discretization errors on regular smooth grids. However, it is often misleading in application to finite-volume discretization schemes on irregular (e.g., unstructured) grids. Convergence of truncation errors severely degrades on general irregular grids; a design-order convergence can be achieved only on grids with a certain degree of geometric regularity. Such degradation of truncation-error convergence does not necessarily imply a lower-order convergence of discretization errors. In these notes, irregular-grid computations demonstrate that the design-order discretization-error convergence can be achieved even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all.

Diskin, Boris↗
DOE OSTI2022

LaplaceInterpolation.jl: A Julia package for fast interpolation on a grid

We implement a linear-time algorithm for interpolation on a regular multidimensional grid in the Julia language. The algorithm is an approximate Laplace interpolation (Press, 1992) when no parameters are given; and when parameters m ∈ Z and ϵ > 0 are set, the interpolant approximates a Matérn kernel, of which radial basis functions and polyharmonic splines are a special case. We implement, in addition, Neumann, Dirichlet (trivial), and average boundary conditions with potentially different aspect ratios in the different dimensions. The interpolant functions in arbitrary dimensions.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1990

Pyramidal Image-Processing Code For Hexagonal Grid

Algorithm based on processing of information on intensities of picture elements arranged in regular hexagonal grid. Called "image pyramid" because image information at each processing level arranged in hexagonal grid having one-seventh number of picture elements of next lower processing level, each picture element derived from hexagonal set of seven nearest-neighbor picture elements in next lower level. At lowest level, fine-resolution of elements of original image. Designed to have some properties of image-coding scheme of primate visual cortex.

Watson, Andrew B.↗
DOE OSTI2024

The Scientific Impact of the Exascale Computing Project

The recent arrival of the Frontier Supercomputer at Oak Ridge National Laboratory officially marked the dawn of the exascale computing era. Its successful deployment coincided with the culmination of the U.S. Department of Energy Exascale Computing Project (ECP), an ambitious, complex, and risky research and development effort that integrated contributions from a broad and diverse subset of the high-performance computing community. The success of ECP will ultimately be judged by the scientific and engineering advances that it enabled. In conclusion, this Special Issue is focused on showcasing early successes in the use of exascale resources to enable breakthroughs in key areas of science in engineering.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1997

Tools for Analysis and Visualization of Large Time-Varying CFD Data Sets

In the second year, we continued to built upon and improve our scanline-based direct volume renderer that we developed in the first year of this grant. This extremely general rendering approach can handle regular or irregular grids, including overlapping multiple grids, and polygon mesh surfaces. It runs in parallel on multi-processors. It can also be used in conjunction with a k-d tree hierarchy, where approximate models and error terms are stored in the nodes of the tree, and approximate fast renderings can be created. We have extended our software to handle time-varying data where the data changes but the grid does not. We are now working on extending it to handle more general time-varying data. We have also developed a new extension of our direct volume renderer that uses automatic decimation of the 3D grid, as opposed to an explicit hierarchy. We explored this alternative approach as being more appropriate for very large data sets, where the extra expense of a tree may be unacceptable. We also describe a new approach to direct volume rendering using hardware 3D textures and incorporates lighting effects. Volume rendering using hardware 3D textures is extremely fast, and machines capable of using this technique are becoming more moderately priced. While this technique, at present, is limited to use with regular grids, we are pursuing possible algorithms extending the approach to more general grid types. We have also begun to explore a new method for determining the accuracy of approximate models based on the light field method described at ACM SIGGRAPH '96. In our initial implementation, we automatically image the volume from 32 equi-distant positions on the surface of an enclosing tessellated sphere. We then calculate differences between these images under different conditions of volume approximation or decimation. We are studying whether this will give a quantitative measure of the effects of approximation. We have created new tools for exploring the differences between images produced by various rendering methods. Images created by our software can be stored in the SGI RGB format. Our idtools software reads in pair of images and compares them using various metrics. The differences of the images using the RGB, HSV, and HSL color models can be calculated and shown. We can also calculate the auto-correlation function and the Fourier transform of the image and image differences. We will explore how these image differences compare in order to find useful metrics for quantifying the success of various visualization approaches. In general, progress was consistent with our research plan for the second year of the grant.

Wilhelms, Jane↗
DOE OSTI2025

ILAMBv2.7 benchmarking results comparing E3SMv2.1 land-atmosphere coupled (BGCv2LNDATM) and stand alone land (ELM) simulations with CMIP6 emission driven historical simulations

This dataset contains land model benchmarking results for the Energy Exascale Earth System Model version 2.1 (E3SMv2.1), including outputs from both coupled biogeochemistry simulations and stand-alone land model simulations. These results are compared against several emission-driven historical simulations from the Coupled Model Intercomparison Project Phase 6 (CMIP6). Benchmarking was conducted using the International Land Model Benchmarking (ILAMB) package, version 2.7 (ILAMBv2.7). CMIP6 model outputs were sourced from the Earth System Grid Federation (ESGF), while the E3SMv2.1 results were derived from raw model outputs. These outputs underwent processing steps such as time serialization, conservative regridding, and data standardization to ensure comparability. For spatial interpolation, the Earth System Modeling Framework (ESMF) tool, ESMF_RegridWeightGen, was employed to generate regridding weights, enabling the transformation of E3SM’s native cubed-sphere grid to a regular latitude-longitude grid.

Feng, Sha [PNNL]↗
NASA NTRS2009

Conical-Domain Model for Estimating GPS Ionospheric Delays

The conical-domain model is a computational model, now undergoing development, for estimating ionospheric delays of Global Positioning System (GPS) signals. Relative to the standard ionospheric delay model described below, the conical-domain model offers improved accuracy. In the absence of selective availability, the ionosphere is the largest source of error for single-frequency users of GPS. Because ionospheric signal delays contribute to errors in GPS position and time measurements, satellite-based augmentation systems (SBASs) have been designed to estimate these delays and broadcast corrections. Several national and international SBASs are currently in various stages of development to enhance the integrity and accuracy of GPS measurements for airline navigation. In the Wide Area Augmentation System (WAAS) of the United States, slant ionospheric delay errors and confidence bounds are derived from estimates of vertical ionospheric delay modeled on a grid at regularly spaced intervals of latitude and longitude. The estimate of vertical delay at each ionospheric grid point (IGP) is calculated from a planar fit of neighboring slant delay measurements, projected to vertical using a standard, thin-shell model of the ionosphere. Interpolation on the WAAS grid enables estimation of the vertical delay at the ionospheric pierce point (IPP) corresponding to any arbitrary measurement of a user. (The IPP of a given user s measurement is the point where the GPS signal ray path intersects a reference ionospheric height.) The product of the interpolated value and the user s thin-shell obliquity factor provides an estimate of the user s ionospheric slant delay. Two types of error that restrict the accuracy of the thin-shell model are absent in the conical domain model: (1) error due to the implicit assumption that the electron density is independent of the azimuthal angle at the IPP and (2) error arising from the slant-to-vertical conversion. At low latitudes or at mid-latitudes under disturbed conditions, the accuracy of SBAS systems based upon the thin-shell model suffers due to the presence of complex ionospheric structure, high delay values, and large electron density gradients. Interpolation on the vertical delay grid serves as an additional source of delay error. The conical-domain model permits direct computation of the user s slant delay estimate without the intervening use of a vertical delay grid. The key is to restrict each fit of GPS measurements to a spatial domain encompassing signals from only one satellite. The conical domain model is so named because each fit involves a group of GPS receivers that all receive signals from the same GPS satellite (see figure); the receiver and satellite positions define a cone, the satellite position being the vertex. A user within a given cone evaluates the delay to the satellite directly, using (1) the IPP coordinates of the line of sight to the satellite and (2) broadcast fit parameters associated with the cone. The conical-domain model partly resembles the thin-shell model in that both models reduce an inherently four-dimensional problem to two dimensions. However, unlike the thin-shell model, the conical domain model does not involve any potentially erroneous simplifying assumptions about the structure of the ionosphere. In the conical domain model, the initially four-dimensional problem becomes truly two-dimensional in the sense that once a satellite location has been specified, any signal path emanating from a satellite can be identified by only two coordinates; for example, the IPP coordinates. As a consequence, a user s slant-delay estimate converges to the correct value in the limit that the receivers converge to the user s location (or, equivalently, in the limit that the measurement IPPs converge to the user s IPP).

Sparks, Lawrence↗
NASA NTRS2004

Partitioning a Gridded Rectangle into Smaller Rectangles

A relatively simple algorithm, devised for use in an image-data-compression application, partitions a rectangular pixelated image (or any other rectangle on which a regular rectangular grid has already been superimposed) into a specified number of smaller rectangles, hereafter denoted segments. The algorithm has the following properties: No floating-point operations are needed. The segments tend to be nearly square (in the sense that their widths and heights in pixel units tend to be nearly equal). The segments tend to have nearly equal areas. The algorithm yields valid results (no zero-width or zero-height segments) as long as the specified number of segments, s, does not exceed the number of pixels (equivalently, the number of grid cells). The inputs to the algorithm are the positive integer s plus the positive integers h and w, denoting the height and width, respectively, of the rectangle in pixel units. The limit on s for a valid result is given by s less than or equal to wh.

Klimesh, Matthew↗
DOE OSTI2025

A Block-Structured Adaptive Mesh Framework to Solve Radiation Transfer Equation in Irregular Embedded Geometries

Radiation transport arises in various scientific, industrial, and medical fields, and understanding its effect in applications is needed to make accurate predictions, safety assessments and performance optimizations. Solving the Radiation Transport Equation (RTE) is challenging due to its integro-differential nature, which involves both differential and integral terms. The differential term describes the change in radiation intensity due to absorption and emission, while the integral term accounts for scattering. The accurate modeling of radiation is further complicated in many applications due to the complex, irregular geometries. Various methods exist for solving the RTE, including the zonal, Monte Carlo, spherical harmonics, discrete ordinates, and finite volume methods. Traditional mesh-based approaches, which rely on structured or unstructured meshes, struggle with irregular geometries due to: a) the difficulty of conforming structured grids to irregular domains, b) challenges in enforcing boundary conditions correctly, and c) the additional computational cost of unstructured mesh methods. This work presents a second-order accurate method for solving the RTE in irregular geometries. The radiation intensity is discretized using the finite-volume method in both spatial and angular directions on regular Cartesian grid blocks. Leveraging the block-structured adaptive mesh refinement (AMR) framework provided by AMReX, our method refines the grid locally to reduce spatial discretization error, ensuring a converged numerical solution while minimizing computational costs elsewhere. A two-stage deferred correction approach is employed: First, a first-order discretization on grid blocks is solved using an algebraic multigrid method in HYPRE. Second, a correction term is applied explicitly to achieve second-order accuracy. The correction term is calculated by approximating the radiation flux on cell faces using a Total Variation Diminishing (TVD) scheme. This approach ensures quick convergence of the multigrid method while preserving higher-order accuracy of the numerical solution. Irregular geometries are resolved as embedded boundaries (EB), resulting in both cut cells and regular cells. In cut cells, we modify the fluxes using face fractions and incorporate additional contributions from EB boundary conditions. To ensure higher-order convergence near the EB interface, the correction term is modified by interpolating the radiation intensity to fictitious ghost points. The implementation takes advantage of modern supercomputers by leveraging AMReX’sMPI/X parallelization strategy where X can be MPI or a GPU accelerator including CUDA, HIP and DPC++. We validate our solver using classical test cases, both with and without EB, demonstrating accuracy and efficiency. Additionally, we analyze the impact of adaptive mesh refinement on solution accuracy and computational cost, highlighting the advantages of our approach for high-resolution radiation transport simulations.

computational fluid dynamics (CFD)↗
NASA NTRS2024

High-Order/Low-Dissipation Chain-Rule Flux Solution Reconstruction Schemes in FUN3D

In this paper, we report progress in the development of economically high-order flux-solution-reconstruction (FSR) schemes, which are second-order accurate on general unstructured grids but achieve high-order accuracy when a grid is regular, i.e., has the same stencil (with the same spacing) throughout the domain. Two variants of the FSR schemes are discussed: chain-rule-flux-solution reconstruction (CFSR) and quadratic-form-flux-solution reconstruction (QFSR), where the former is based on the chain rule and the latter on the flux reconstruction expressed as a function of solution variables. These schemes are tested for flows with shock waves with a limiter incorporated in the flux and solution reconstructions. Improved results, compared with second-order methods are demonstrated for inviscid and viscous flows with smooth grids.

high-order↗
DOE OSTI2026

Fast solvers for tokamak fluid models with PETSc

Multigrid (MG) is widely recognized as a highly effective solver for the model problem, the Laplacian, but textbook MG fails on most problems of interest. MG methods have been applied to complex, real-world applications with careful consideration of the physical model and discretization. In this work we develop the first step in applying MG methods to science and engineering relevant magnetohydrodynamics (MHD) tokamak models in the M3D-C1 (https://m3dc1.pppl.gov) fusion energy science code. The semi-implicit time integrator in M3D-C1 is composed of many linear solves. The implicit advance of the momentum equation is the most challenging and is the focus of this work. The current production solver in M3D-C1 is a block Jacobi (BJ) preconditioner within a Krylov solver, where blocks group degrees of freedom on planes of constant toroidal coordinate. BJ convergence degrades as the number of planes increases due to the spectral properties of the matrix preconditioned with BJ. The partially magnetic field-aligned, regular toroidal grid structure in M3D-C1 is amenable to semi-coarsening geometric MG in the toroidal direction. This paper develops such a solver and demonstrates competitive performance on a runaway electron model of a SPARC (https://cfs.energy/technology/sparc) disruption, and superior robustness on a stellarator model on which the BJ solver fails to converge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

High‐speed 4‐dimensional scanning transmission electron microscopy using compressive sensing techniques

Abstract Here we show that compressive sensing allows 4‐dimensional (4‐D) STEM data to be obtained and accurately reconstructed with both high‐speed and reduced electron fluence. The methodology needed to achieve these results compared to conventional 4‐D approaches requires only that a random subset of probe locations is acquired from the typical regular scanning grid, which immediately generates both higher speed and the lower fluence experimentally. We also consider downsampling of the detector, showing that oversampling is inherent within convergent beam electron diffraction (CBED) patterns and that detector downsampling does not reduce precision but allows faster experimental data acquisition. Analysis of an experimental atomic resolution yttrium silicide dataset shows that it is possible to recover over 25 dB peak signal‐to‐noise ratio in the recovered phase using 0.3% of the total data. Lay abstract : Four‐dimensional scanning transmission electron microscopy (4‐D STEM) is a powerful technique for characterizing complex nanoscale structures. In this method, a convergent beam electron diffraction pattern (CBED) is acquired at each probe location during the scan of the sample. This means that a 2‐dimensional signal is acquired at each 2‐D probe location, equating to a 4‐D dataset. Despite the recent development of fast direct electron detectors, some capable of 100kHz frame rates, the limiting factor for 4‐D STEM is acquisition times in the majority of cases, where cameras will typically operate on the order of 2kHz. This means that a raster scan containing 256^2 probe locations can take on the order of 30s, approximately 100‐1000 times longer than a conventional STEM imaging technique using monolithic radial detectors. As a result, 4‐D STEM acquisitions can be subject to adverse effects such as drift, beam damage, and sample contamination. Recent advances in computational imaging techniques for STEM have allowed for faster acquisition speeds by way of acquiring only a random subset of probe locations from the field of view. By doing this, the acquisition time is significantly reduced, in some cases by a factor of 10‐100 times. The acquired data is then processed to fill‐in or inpaint the missing data, taking advantage of the inherently low‐complex signals which can be linearly combined to recover the information. In this work, similar methods are demonstrated for the acquisition of 4‐D STEM data, where only a random subset of CBED patterns are acquired over the raster scan. We simulate the compressive sensing acquisition method for 4‐D STEM and present our findings for a variety of analysis techniques such as ptychography and differential phase contrast. Our results show that acquisition times can be significantly reduced on the order of 100‐300 times, therefore improving existing frame rates, as well as further reducing the electron fluence beyond just using a faster camera.

Robinson, Alex W.↗
DOE OSTI2020

miniWeather

The code “miniWeather” solves the 2-D Euler equations with a Finite-Volume method on a regular Cartesian grid with Runge-Kutta time stepping, 4th-order hypervisocisty, and 4th-order spatial accuracy. The code is primarily a parallel programming training mini application in Fortran, C, and C++ with MPI, OpenMP, OpenMP offload, OpenACC, and C++ portability approaches. It can also serve as a mini application for acceptance testing for new machines and compilers for both functionality and performance.

Norman, MatthewR↗
DOE OSTI2026

RHOD Site - NOAA PSL Wind Retrievals WINDoe / Derived Data

This dataset contains daily NetCDF files with horizontal wind profiles retrieved with the WINDoe retrieval (Gebauer and Bell 2024) at Rhode Island (RHOD). WINDoe retrievals datasets are also available at Nantucket Island (NANT, nant.windoe.z01.c1) and Block Island (BLOC, bloc.windoe.z01.c1). WINDoe is an optimal estimation algorithm to retrieve wind profiles combining multiple instruments. The code is available in this github repository (https://github.com/OAR-atmospheric-observations/WINDoe/tree/main) and the retrieval is described by Gebauer and Bell (2024). WINDoe allows combining the individual datasets and outputs into one profile taking into account the information and uncertainties of each dataset. The use of WINDoe minimizes data gaps and maximizes data availability, compared to using wind profiles from only one of the instruments. The regular height grid eases comparisons to numerical weather prediction models. Code modifications have been made that include reading in WFIP3 specific instruments, averaging Doppler lidar radial velocities at various azimuth angles to avoid overfitting, and allowing the user to define a height grid by the user in the vipfile. The instruments used as input to the retrieval are a radar wind profiler (low- and high resolution mode) providing data in and above the boundary layer, a scanning Doppler lidar usually providing data throughout the boundary layer, a profiling lidar providing data from 50 to 200 m at BLOC and NANT, and from 10 to 280 m at Rhode Island, and a surface tower (4 m at NANT and RHOD and 10 m at BLOC). From the scanning lidars, we used radial velocity measurements at 60 deg elevation angle at six different azimuth angles with a resolution of approximately 30 m along the line of sight and the lowest range gate at approximately 70 m. The wind profiles are retrieved with WINDoe up to 3.74 km with 10 m vertical resolution. The profiles are retrieved every 15 min at BLOC and NANT and every 60 min at RHOD.

17 WIND ENERGY↗
DOE OSTI2026

BLOC Site - NOAA PSL Wind Retrievals WINDoe / Derived Data

This dataset contains daily netcdf files with horizontal wind profiles retrieved with the WINDoe retrieval (Gebauer and Bell 2024) at Block Island (BLOC). WINDoe retrievals datasets are also available at Nantucket Island (NANT, nant.windoe.z01.c1) and Rhode Island (RHOD, rhod.windoe.z01.c1). WINDoe is an optimal estimation algorithm to retrieve wind profiles combining multiple instruments. The code is available in this github repository (https://github.com/OAR-atmospheric-observations/WINDoe/tree/main), and the retrieval is described by Gebauer and Bell (2024). WINDoe allows combining the individual datasets and outputs into one profile taking into account the information and uncertainties of each dataset. The use of WINDoe minimizes data gaps and maximizes data availability, compared to using wind profiles from only one of the instruments. The regular height grid eases comparisons to numerical weather prediction models. Code modifications have been made that include reading in WFIP3 specific instruments, averaging Doppler lidar radial velocities at various azimuth angles to avoid overfitting, and allowing the user to define a height grid by the user in the vipfile. The instruments used as input to the retrieval are a radar wind profiler (low- and high resolution mode) providing data in and above the boundary layer, a scanning Doppler lidar usually providing data throughout the boundary layer, a profiling lidar providing data from 50 to 200 m at BLOC and NANT, and from 10 to 280 m at Rhode Island, and a surface tower (4 m at NANT and RHOD and 10 m at BLOC). From the scanning lidars, we used radial velocity measurements at 60 deg elevation angle at six different azimuth angles with a resolution of approximately 30 m along the line of sight and the lowest range gate at approximately 70 m. The wind profiles are retrieved with WINDoe up to 3.74 km with 10 m vertical resolution. The profiles are retrieved every 15 min at BLOC and NANT and every 60 min at RHOD.

17 WIND ENERGY↗
DOE OSTI2026

NANT Site - NOAA PSL Wind Retrievals WINDoe / Derived Data

This dataset contains daily NetCDF files with horizontal wind profiles retrieved with the WINDoe retrieval (Gebauer and Bell 2024) at Nantucket Island (NANT). WINDoe retrievals datasets are also available at Block Island (BLOC, bloc.windoe.z01.c1) and Rhode Island (RHOD, rhod.windoe.z01.c1). WINDoe is an optimal estimation algorithm to retrieve wind profiles combining multiple instruments. The code is available in this github repository (https://github.com/OAR-atmospheric-observations/WINDoe/tree/main), and the retrieval is described by Gebauer and Bell (2024). WINDoe allows combining the individual datasets and outputs into one profile taking into account the information and uncertainties of each dataset. The use of WINDoe minimizes data gaps and maximizes data availability, compared to using wind profiles from only one of the instruments. The regular height grid eases comparisons to numerical weather prediction models. Code modifications have been made that include reading in WFIP3 specific instruments, averaging Doppler lidar radial velocities at various azimuth angles to avoid overfitting, and allowing the user to define a height grid by the user in the vipfile. The instruments used as input to the retrieval are a radar wind profiler (low- and high resolution mode) providing data in and above the boundary layer, a scanning Doppler lidar usually providing data throughout the boundary layer, a profiling lidar providing data from 50 to 200 m at BLOC and NANT, and from 10 to 280 m at Rhode Island, and a surface tower (4 m at NANT and RHOD and 10 m at BLOC). From the scanning lidars, we used radial velocity measurements at 60 deg elevation angle at six different azimuth angles with a resolution of approximately 30 m along the line of sight and the lowest range gate at approximately 70 m. The wind profiles are retrieved with WINDoe up to 3.74 km with 10 m vertical resolution. The profiles are retrieved every 15 min at BLOC and NANT and every 60 min at RHOD.

17 WIND ENERGY↗
NASA NTRS1985

Balloon-based interferometric techniques

A balloon-borne triple-etalon Fabry-Perot Interferometer, observing the Doppler shifts of absorption lines caused by molecular oxygen and water vapor in the far red/near infrared spectrum of backscattered sunlight, has been used to evaluate a passive spaceborne remote sensing technique for measuring winds in the troposphere and stratosphere. There have been two successful high altitude balloon flights of the prototype UCL instrument from the National Scientific Balloon Facility at Palestine, TE (May 80, Oct. 83). The results from these flights have demonstrated that an interferometer with adequate resolution, stability and sensitivity can be built. The wind data are of comparable quality to those obtained from operational techniques (balloon and rocket sonde, cloud-top drift analysis, and from the gradient wind analysis of satellite radiance measurements). However, the interferometric data can provide a regular global grid, over a height range from 5 to 50 km in regions of clear air. Between the middle troposphere (5 km) and the upper stratosphere (40 to 50 km), an optimized instrument can make wind measurements over the daylit hemisphere with an accuracy of about 3 to 5 m/sec (2 sigma). It is possible to obtain full height profiles between altitudes of 5 and 50 km, with 4 km height resolution, and a spatial resolution of about 200 km, along the orbit track. Below an altitude of about 10 km, Fraunhofer lines of solar origin are possible targets of the Doppler wind analysis. Above an altitude of 50 km, the weakness of the backscattered solar spectrum (decreasing air density) is coupled with the low absorption crosssection of all atmospheric species in the spectral region up to 800 nm (where imaging photon detectors can be used), causing the along-the-track resolution (or error) to increase beyond values useful for operational purposes. Within the region of optimum performance (5 to 50 km), however, the technique is a valuable potential complement to existing wind measuring systems and can provide a low cost addition to powerful active (LIDAR) wind measuring systems now under development.

Rees, David↗
NASA NTRS1993

Multigrid properties of upwind-biased data reconstructions

The multigrid properties of two data reconstruction methods used for achieving second-order spatial accuracy when solving the two-dimensional Euler equations are examined. The data reconstruction methods are used with an implicit upwind algorithm which uses linearized backward-Euler time-differencing. The solution of the resulting linear system is performed by an iterative procedure. In the present study only regular quadrilateral grids are considered, so a red-black Gauss-Seidel iteration is used. Although the Jacobian is approximated by first-order upwind extrapolation, two alternative data reconstruction techniques for the flux integral that yield higher-order spatial accuracy at steady state are examined. The first method, probably most popular for structured quadrilateral grids, is based on estimating the cell gradients using one-dimensional reconstruction along curvilinear coordinates. The second method is based on Green's theorem. Analysis and numerical results for the two dimensional Euler equations show that data reconstruction based on Green's theorem has superior multigrid properties as compared to the one-dimensional data reconstruction method.

Warren, Gary P.↗