SEARCH · Engineering Papers

Results for “spatial interpolation”

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 343 records · Page 19

All sources NASA NTRS DOE OSTI

DOE OSTI2017

Dual gauge field theory of quantum liquid crystals in two dimensions

We present a self-contained review of the theory of dislocation-mediated quantum melting at zero temperature in two spatial dimensions. The theory describes the liquid-crystalline phases with spatial symmetries in between a quantum crystalline solid and an isotropic superfluid: quantum nematics and smectics. It is based on an Abelian-Higgs-type duality mapping of phonons onto gauge bosons (“stress photons”), which encode for the capacity of the crystal to propagate stresses. Dislocations and disclinations, the topological defects of the crystal, are sources for the gauge fields and the melting of the crystal can be understood as the proliferation (condensation) of these defects, giving rise to the Anderson–Higgs mechanism on the dual side. For the liquid crystal phases, the shear sector of the gauge bosons becomes massive signaling that shear rigidity is lost. After providing the necessary background knowledge, including the order parameter theory of two-dimensional quantum liquid crystals and the dual theory of stress gauge bosons in bosonic crystals, the theory of melting is developed step-by-step via the disorder theory of dislocation-mediated melting. Resting on symmetry principles, we derive the phenomenological imaginary time actions of quantum nematics and smectics and analyze the full spectrum of collective modes. The quantum nematic is a superfluid having a true rotational Goldstone mode due to rotational symmetry breaking, and the origin of this ‘deconfined’ mode is traced back to the crystalline phase. The two-dimensional quantum smectic turns out to be a dizzyingly anisotropic phase with the collective modes interpolating between the solid and nematic in a non-trivial way. We also consider electrically charged bosonic crystals and liquid crystals, and carefully analyze the electromagnetic response of the quantum liquid crystal phases. In particular, the quantum nematic is a real superconductor and shows the Meissner effect. Furthermore, their special properties inherited from spatial symmetry breaking show up mostly at finite momentum, and should be accessible by momentum-sensitive spectroscopy.

36 MATERIALS SCIENCE↗

DOE OSTI2011

acousto-electricV1.0

Seismo-electric for both seismic and electric sources Code is for a layered acoustic and electric medium. Seismic calculated at a time step delta_t, the electric and magnetic fields calculated at a smaller time step within delta_t. Seismic calculated on a smaller grid than the EM calculations. At appropriate time steps pressure is interpolated onto the EM grid and/or electric fields interpolated onto the seismic grid. This version takes into the air-earth interface for both seismic and EM wave fields. Solutions of coupled first order Maxwell's equations are solved in the two dimensional environment using a finite- difference scheme on a staggered spatial and temporal grid. The acoustic equations (where shear is ignored) are obtained from the poroelastic governing equations. The acoustic equations are solved using a finite-difference approach with a staggered grid in space and time.

Bartel, Lewis C.↗

DOE OSTI2016

Explicit modeling of abiotic and landscape factors reveals precipitation and forests associated with aphid abundance

Abstract Increases in natural or noncrop habitat surrounding agricultural fields have been shown to be correlated with declines in insect crop pests. However, these patterns are highly variable across studies suggesting other important factors, such as abiotic drivers, which are rarely included in landscape models, may also contribute to variability in insect population abundance. The objective of this study was to explicitly account for the contribution of temperature and precipitation, in addition to landscape composition, on the abundance of a widespread insect crop pest, the soybean aphid ( Aphis glycines Matsumura), in Wisconsin soybean fields. We hypothesized that higher soybean aphid abundance would be associated with higher heat accumulation (e.g., growing degree days) and increasing noncrop habitat in the surrounding landscape, due to the presence of the overwintering primary hosts of soybean aphid. To evaluate these hypotheses, we used an ecoinformatics approach that relied on a large dataset collected across Wisconsin over a 9‐year period (2003–2011), for an average of 235 sites per year ( n = 2,110 fields total). We determined surrounding landscape composition (1.5‐km radius) using publicly available satellite‐derived land cover imagery and interpolated daily temperature and precipitation information from the National Weather Service COOP weather station network. We constructed linear mixed models for soybean aphid abundance based on abiotic and landscape explanatory variables and applied model averaging for prediction using an information theoretic framework. Over this broad spatial and temporal extent in Wisconsin, we found that variation in growing season precipitation was positively related to soybean aphid abundance, while higher precipitation during the nongrowing season had a negative effect on aphid populations. Additionally, we found that aphid populations were higher in areas with proportionally more forest but were lower in areas where minor crops, such as small grains, were more prevalent. Thus, our findings support our hypothesis that including abiotic drivers increases our understanding of crop pest abundance and distribution. Moreover, by explicitly modeling abiotic factors, we may be able to explore how variable climate in tandem with land cover patterns may affect current and future insect populations, with potentially critical implications for crop yields and agricultural food webs.

Stack Whitney, Kaitlin↗

DOE OSTI2016

Hybrid PN-SN Calculations with SAAF for the Multiscale Transport Capability in Rattlesnake

Two interface conditions, the Lagrange multiplier method and the upwinding method, for hybrid \pn-\sn calculations is proposed for the self-adjoint angular flux (SAAF) formulation of the transport equation using the continuous finite element method (FEM) for spatial discretization. These interface conditions are implemented in Rattlesnake, the radiation transport application built on MOOSE, for the on-going multiscale transport simulation effort at INL. For smoothing the solution at the interface for the Lagrange multiplier method, a method based on \sn Lagrange interpolation on the sphere is proposed. Numerical results indicate that the interface conditions give the expected convergence.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Characterizing the Spatial Scales of Extreme Daily Precipitation in the United States

The spatial extent of an extreme precipitation event can be important for a basin’s hydrologic response and subsequent flood risk, and may yield insights into underlying atmospheric processes. Using a relaxed moving-neighborhood approach, we develop indicator semivariograms based on precipitation records from the Global Historical Climatology Network–Daily (GHCN-D) station network to directly quantify the climatological length scales of extreme daily precipitation over the United States during 1965–2014. We find that the length scales of extreme (90th percentile) daily precipitation events vary both regionally and seasonally. Over the eastern half of the United States, daily extreme precipitation length scales reach 400 km during the winter months, but are approximately half as large during the summer months. The Northwest region, on the other hand, exhibits little seasonal variation, with extreme precipitation length scales of approximately 150 km throughout the year. By leveraging in situ station measurements, our study avoids some of the uncertainties associated with satellite or interpolated precipitation data, and provides the longest climatological assessment of length scales of extreme daily precipitation over the United States to date. Although the length scales that we calculate can be sensitive to station density, neighborhood size, and neighborhood relaxation, we find that the interregional and interseasonal differences in length scales are relatively robust. Our method could be extended to quantify changes in the spatial extent of extreme daily precipitation in the recent past, and to investigate the underlying causes of any changes that are detected.

Touma, Danielle↗

DOE OSTI2014

Efficient modeling of photonic crystals with local Hermite polynomials

Developing compact algorithms for accurate electrodynamic calculations with minimal computational cost is an active area of research given the increasing complexity in the design of electromagnetic composite structures such as photonic crystals, metamaterials, optical interconnects, and on-chip routing. We show that electric and magnetic (EM) fields can be calculated using scalar Hermite interpolation polynomials as the numerical basis functions without having to invoke edge-based vector finite elements to suppress spurious solutions or to satisfy boundary conditions. This approach offers several fundamental advantages as evidenced through band structure solutions for periodic systems and through waveguide analysis. Compared with reciprocal space (plane wave expansion) methods for periodic systems, advantages are shown in computational costs, the ability to capture spatial complexity in the dielectric distributions, the demonstration of numerical convergence with scaling, and variational eigenfunctions free of numerical artifacts that arise from mixed-order real space basis sets or the inherent aberrations from transforming reciprocal space solutions of finite expansions. The photonic band structure of a simple crystal is used as a benchmark comparison and the ability to capture the effects of spatially complex dielectric distributions is treated using a complex pattern with highly irregular features that would stress spatial transform limits. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Biases in southern hemisphere climate trends induced by coarsely specifying the temporal resolution of stratospheric ozone

Abstract Global climate models that do not include interactive middle atmosphere chemistry, such as most of those contributing to the Coupled Model Intercomparison Project Phase 5, typically specify stratospheric ozone using monthly mean, zonal mean values and linearly interpolate to the time resolution of the model. We show that this method leads to significant biases in the simulated climate of the southern hemisphere (SH) over the late twentieth century. Previous studies have attributed similar biases in simulated SH climate change to the effect of the spatial smoothing of the specified ozone, i.e., to using zonal mean concentrations. We here show that the bias in climate trends due to undersampling of the rapid temporal changes in ozone during the seasonal evolution of the Antarctic ozone hole is considerable and reaches all the way into the troposphere. Our results suggest that the bias can be substantially reduced by specifying daily ozone concentrations.

Neely, III, R. R.↗

DOE OSTI2019

A cloud-free MODIS snow cover dataset for the contiguous United States from 2000 to 2017

This article presents a cloud-free snow cover dataset with a daily temporal resolution and 0.05° spatial resolution from March 2000 to February 2017 over the contiguous United States (CONUS). The dataset was developed by completely removing clouds from the original NASA’s Moderate Resolution Imaging Spectroradiometer (MODIS) Snow Cover Area product (MOD10C1) through a series of spatiotemporal filters followed by the Variational Interpolation (VI) algorithm; the filters and VI algorithm were evaluated using bootstrapping test. The dataset was validated over the period with the Landsat 7 ETM+ snow cover maps in the Seattle, Minneapolis, Rocky Mountains, and Sierra Nevada regions. The resulting cloud-free snow cover captured accurately dynamic changes of snow throughout the period in terms of Probability of Detection (POD) and False Alarm Ratio (FAR) with average values of 0.955 and 0.179 for POD and FAR, respectively. The dataset provides continuous inputs of snow cover area for hydrologic studies for almost two decades. The VI algorithm can be applied in other regions given that a proper validation can be performed.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

A Scalable Gaussian Process Approach to Shear Mapping with MuyGPs

Analysis of cosmic shear is an integral part of understanding structure growth across cosmic time, which in turn provides us with information about the nature of dark energy. Conventional methods generate shear maps from which we can infer the matter distribution in the universe. Current methods (e.g., Kaiser–Squires inversion) for generating these maps, however, are tricky to implement and can introduce bias. Recent alternatives construct a spatial process prior for the lensing potential, which allows for inference of the convergence and shear parameters given lensing shear measurements. Realizing these spatial processes, however, scales cubically in the number of observations—an unacceptable expense as near-term surveys expect billions of correlated measurements. Therefore, we present a linearly scaling shear map construction alternative using a scalable Gaussian process prior called MuyGPs. MuyGPs avoids cubic scaling by conditioning interpolation on only nearest neighbors and fits hyperparameters using batched leave-one-out cross-validation. This work is the first step toward a full, scalable mass mapping method. We work in a simplified regime where we validate our method by interpolating and analyzing maps given noisy point-estimate data from all three shear fields, taken from a suite of N -body ray-tracing simulations. We also show that we can perform these operations at the scale of billions of galaxies on high-performance computing platforms.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1987

Image reconstruction from multiple 1-D scans using filtered localized projection

The spatial resolution that can be attained using scanning linear arrays (consisting of discrete IR solid-state detectors) for image acquisition is considered, and a filtered local projection (FLP) method is described which efficiently combines all available scan information into one rectangular grid without the need for explicit interpolation. Mathematically, the FLP algorithm consists of a localized summation followed by an inverse-filter operation, and it has application to nonlinear restoring techniques. The present method is applied, using a linear array, to simulated data for staggered parallel scans and to multiple scan directions. Noise effects and limitations of the technique are also considered.

Frieden, B. Roy↗

NASA NTRS2001

Error Estimation of An Ensemble Statistical Seasonal Precipitation Prediction Model

This NASA Technical Memorandum describes an optimal ensemble canonical correlation forecasting model for seasonal precipitation. Each individual forecast is based on the canonical correlation analysis (CCA) in the spectral spaces whose bases are empirical orthogonal functions (EOF). The optimal weights in the ensemble forecasting crucially depend on the mean square error of each individual forecast. An estimate of the mean square error of a CCA prediction is made also using the spectral method. The error is decomposed onto EOFs of the predictand and decreases linearly according to the correlation between the predictor and predictand. Since new CCA scheme is derived for continuous fields of predictor and predictand, an area-factor is automatically included. Thus our model is an improvement of the spectral CCA scheme of Barnett and Preisendorfer. The improvements include (1) the use of area-factor, (2) the estimation of prediction error, and (3) the optimal ensemble of multiple forecasts. The new CCA model is applied to the seasonal forecasting of the United States (US) precipitation field. The predictor is the sea surface temperature (SST). The US Climate Prediction Center's reconstructed SST is used as the predictor's historical data. The US National Center for Environmental Prediction's optimally interpolated precipitation (1951-2000) is used as the predictand's historical data. Our forecast experiments show that the new ensemble canonical correlation scheme renders a reasonable forecasting skill. For example, when using September-October-November SST to predict the next season December-January-February precipitation, the spatial pattern correlation between the observed and predicted are positive in 46 years among the 50 years of experiments. The positive correlations are close to or greater than 0.4 in 29 years, which indicates excellent performance of the forecasting model. The forecasting skill can be further enhanced when several predictors are used.

Shen, Samuel S. P.↗

NASA NTRSDate not supplied

Where IMERG Goes Next: Version 08 and Beyond

With the Version 07 (V07) Integrated Multi-satellitE Retrievals for GPM (IMERG) algorithm finalized and production initiated, the focus turns to enhancements for Version 08. These include innovations not included in V07 due to time constraints, plus issues revealed by the initial V07 products. One high priority is to evaluate and revise the schemes in V07 that rectify temporal artifacts caused by the time interpolation that fills the gaps between the various passive microwave (PMW) sensor overpasses. A second priority is to improve the homogeneity between the TRMM and GPM eras by characterizing differences between the two eras, determining the causes of these differences, and applying corrections as feasible, perhaps by enforcing spatial scale consistency (an overarching issue). Certainly, we must account for GPROF and the Combined Radar-Radiometer Algorithm converting to Machine Learning schemes in V08. Other priority topics include additional automated quality control for artifacts in the IR brightness temperatures and PMW precipitation fields, revisions to the specification algorithm for the probability of liquid precipitation, and accommodating new PMW sensors, which include the next generation of small-sats. We also consider the post-V08 landscape; the final GPM reprocessing will be restricted to fixing known code or algorithmic errors. Nonetheless, there are several data sources on the horizon to consider, including more small-sat PMW radiometers, AVHRR-based precipitation estimates (most useful in high latitudes), and the ISCCP-Next Generation and GEO-Ring projects that could provide easy access to multiple geosynchronous satellite channels and enable significantly improved algorithms compared to GEO-IR alone.

George J. Huffman↗

DOE OSTI2017

Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling

The mass-lumped method avoids the cost of inverting the mass matrix and simultaneously maintains spatial accuracy by adopting additional interior integration points, known as cubature points. To date, such points are only known analytically in tensor domains, such as quadrilateral or hexahedral elements. Thus, the diagonal-mass-matrix spectral element method (SEM) in non-tensor domains always relies on numerically computed interpolation points or quadrature points. However, only the cubature points for degrees 1 to 6 are known, which is the reason that we have developed a p-norm-based optimization algorithm to obtain higher-order cubature points. In this way, we obtain and tabulate new cubature points with all positive integration weights for degrees 7 to 9. The dispersion analysis illustrates that the dispersion relation determined from the new optimized cubature points is comparable to that of the mass and stiffness matrices obtained by exact integration. Simultaneously, the Lebesgue constant for the new optimized cubature points indicates its surprisingly good interpolation properties. As a result, such points provide both good interpolation properties and integration accuracy. The Courant–Friedrichs–Lewy (CFL) numbers are tabulated for the conventional Fekete-based triangular spectral element (TSEM), the TSEM with exact integration, and the optimized cubature-based TSEM (OTSEM). A complementary study demonstrates the spectral convergence of the OTSEM. A numerical example conducted on a half-space model demonstrates that the OTSEM improves the accuracy by approximately one order of magnitude compared to the conventional Fekete-based TSEM. In particular, the accuracy of the 7th-order OTSEM is even higher than that of the 14th-order Fekete-based TSEM. Furthermore, the OTSEM produces a result that can compete in accuracy with the quadrilateral SEM (QSEM). The high accuracy of the OTSEM is also tested with a non-flat topography model. In terms of computational efficiency, the OTSEM is more efficient than the Fekete-based TSEM, although it is slightly costlier than the QSEM when a comparable numerical accuracy is required. - Highlights: • Higher-order cubature points for degrees 7 to 9 are developed. • The effects of quadrature rule on the mass and stiffness matrices has been conducted. • The cubature points have always positive integration weights. • Freeing from the inversion of a wide bandwidth mass matrix. • The accuracy of the TSEM has been improved in about one order of magnitude.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1993

Vision based obstacle detection and grouping for helicopter guidance

Electro-optical sensors can be used to compute range to objects in the flight path of a helicopter. The computation is based on the optical flow/motion at different points in the image. The motion algorithms provide a sparse set of ranges to discrete features in the image sequence as a function of azimuth and elevation. For obstacle avoidance guidance and display purposes, these discrete set of ranges, varying from a few hundreds to several thousands, need to be grouped into sets which correspond to objects in the real world. This paper presents a new method for object segmentation based on clustering the sparse range information provided by motion algorithms together with the spatial relation provided by the static image. The range values are initially grouped into clusters based on depth. Subsequently, the clusters are modified by using the K-means algorithm in the inertial horizontal plane and the minimum spanning tree algorithms in the image plane. The object grouping allows interpolation within a group and enables the creation of dense range maps. Researchers in robotics have used densely scanned sequence of laser range images to build three-dimensional representation of the outside world. Thus, modeling techniques developed for dense range images can be extended to sparse range images. The paper presents object segmentation results for a sequence of flight images.

Sridhar, Banavar↗

NASA NTRS1994

Data assimilation and model evaluation experiment datasets

The Institute for Naval Oceanography, in cooperation with Naval Research Laboratories and universities, executed the Data Assimilation and Model Evaluation Experiment (DAMEE) for the Gulf Stream region during fiscal years 1991-1993. Enormous effort has gone into the preparation of several high-quality and consistent datasets for model initialization and verification. This paper describes the preparation process, the temporal and spatial scopes, the contents, the structure, etc., of these datasets. The goal of DAMEE and the need of data for the four phases of experiment are briefly stated. The preparation of DAMEE datasets consisted of a series of processes: (1) collection of observational data; (2) analysis and interpretation; (3) interpolation using the Optimum Thermal Interpolation System package; (4) quality control and re-analysis; and (5) data archiving and software documentation. The data products from these processes included a time series of 3D fields of temperature and salinity, 2D fields of surface dynamic height and mixed-layer depth, analysis of the Gulf Stream and rings system, and bathythermograph profiles. To date, these are the most detailed and high-quality data for mesoscale ocean modeling, data assimilation, and forecasting research. Feedback from ocean modeling groups who tested this data was incorporated into its refinement. Suggestions for DAMEE data usages include (1) ocean modeling and data assimilation studies, (2) diagnosis and theoretical studies, and (3) comparisons with locally detailed observations.

Lai, Chung-Cheng A.↗

DOE OSTI2015

Theoretical analysis on the measurement errors of local 2D DIC: Part I temporal and spatial uncertainty quantification of displacement measurements

This study presents a theoretical uncertainty quantification of displacement measurements by subset-based 2D-digital image correlation. A generalized solution to estimate the random error of displacement measurement is presented. The obtained solution suggests that the random error of displacement measurements is determined by the image noise, the summation of the intensity gradient in a subset, the subpixel part of displacement, and the interpolation scheme. The proposed method is validated with virtual digital image correlation tests.

2D-DIC↗

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 NTRS2012

Large Eddy Simulation (LES) of Particle-Laden Temporal Mixing Layers

High-fidelity models of plume-regolith interaction are difficult to develop because of the widely disparate flow conditions that exist in this process. The gas in the core of a rocket plume can often be modeled as a time-dependent, high-temperature, turbulent, reacting continuum flow. However, due to the vacuum conditions on the lunar surface, the mean molecular path in the outer parts of the plume is too long for the continuum assumption to remain valid. Molecular methods are better suited to model this region of the flow. Finally, granular and multiphase flow models must be employed to describe the dust and debris that are displaced from the surface, as well as how a crater is formed in the regolith. At present, standard commercial CFD (computational fluid dynamics) software is not capable of coupling each of these flow regimes to provide an accurate representation of this flow process, necessitating the development of custom software. This software solves the fluid-flow-governing equations in an Eulerian framework, coupled with the particle transport equations that are solved in a Lagrangian framework. It uses a fourth-order explicit Runge-Kutta scheme for temporal integration, an eighth-order central finite differencing scheme for spatial discretization. The non-linear terms in the governing equations are recast in cubic skew symmetric form to reduce aliasing error. The second derivative viscous terms are computed using eighth-order narrow stencils that provide better diffusion for the highest resolved wave numbers. A fourth-order Lagrange interpolation procedure is used to obtain gas-phase variable values at the particle locations.

Bellan, Josette↗