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All sourcesNASA NTRSDOE OSTI
DOE OSTI2011

OBTAINING POTENTIAL FIELD SOLUTIONS WITH SPHERICAL HARMONICS AND FINITE DIFFERENCES

Potential magnetic field solutions can be obtained based on the synoptic magnetograms of the Sun. Traditionally, a spherical harmonics decomposition of the magnetogram is used to construct the current- and divergence-free magnetic field solution. This method works reasonably well when the order of spherical harmonics is limited to be small relative to the resolution of the magnetogram, although some artifacts, such as ringing, can arise around sharp features. When the number of spherical harmonics is increased, however, using the raw magnetogram data given on a grid that is uniform in the sine of the latitude coordinate can result in inaccurate and unreliable results, especially in the polar regions close to the Sun. We discuss here two approaches that can mitigate or completely avoid these problems: (1) remeshing the magnetogram onto a grid with uniform resolution in latitude and limiting the highest order of the spherical harmonics to the anti-alias limit; (2) using an iterative finite difference algorithm to solve for the potential field. The naive and the improved numerical solutions are compared for actual magnetograms and the differences are found to be rather dramatic. We made our new Finite Difference Iterative Potential-field Solver (FDIPS) a publicly available code so that other researchers can also use it as an alternative to the spherical harmonics approach.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Iterative diagonalization of the non-Hermitian transcorrelated Hamiltonian using a plane-wave basis set: Application to sp-electron systems with deep core states

We develop an iterative diagonalization scheme in solving a one-body self-consistent-field equation in the transcorrelated (TC) method using a plane-wave basis set. Non-Hermiticity in the TC method is well handled with a block-Davidson algorithm. We verify that the required computational cost is efficiently reduced by our algorithm. In addition, we apply our plane-wave-basis TC calculation to some simple sp-electron systems with deep core states to elucidate an impact of the pseudopotential approximation to the calculated band structures. We find that a position of the deep valence bands is improved by an explicit inclusion of core states, but an overall band structure is consistent with a regular setup that includes core states into the pseudopotentials. This study offers an important understanding for the future application of the TC method to strongly correlated solids.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets: A worked example for ground-based experiments

Context. An estimation of the sky signal from streams of time ordered data (TOD) acquired by the cosmic microwave background (CMB) experiments is one of the most important steps in the context of CMB data analysis referred to as the map-making problem. The continuously growing CMB data sets render the CMB map-making problem progressively more challenging in terms of computational cost and memory in particular in the context of ground-based experiments with their operational limitations as well as the presence of contaminants. Aims. We study a recently proposed, novel class of the Preconditioned Conjugate Gradient (PCG) solvers which invoke two-level preconditioners in the context of the ground-based CMB experiments. We compare them against the PCG solvers commonly used in the map-making context considering their precision and time-to-solution. Methods. We compare these new methods on realistic, simulated data sets reflecting the characteristics of current and forthcoming CMB ground-based experiments. We develop a divide-and-conquer implementation of the approach where each processor performs a sequential map-making for a subset of the TOD. Results. We find that considering the map level residuals, the new class of solvers permits us to achieve a tolerance that is better than the standard approach by up to three orders of magnitude, where the residual level often saturates before convergence is reached. This often corresponds to an important improvement in the precision of the recovered power spectra in particular on the largest angular scales. The new method also typically requires fewer iterations to reach a required precision and therefore shorter run times are required for a single map-making solution. However, the construction of an appropriate two-level preconditioner can be as costly as a single standard map-making run. Nevertheless, if the same problem needs to be solved multiple times, for example, as in Monte Carlo simulations, this cost is incurred only once, and the method should be competitive, not only as far as its precision is concerned but also its performance.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Generating moment matching scenarios using optimization techniques

An optimization based method is proposed to generate moment matching scenarios for numerical integration and its use in stochastic programming. The main advantage of the method is its flexibility: it can generate scenarios matching any prescribed set of moments of the underlying distribution rather than matching all moments up to a certain order, and the distribution can be defined over an arbitrary set. This allows for a reduction in the number of scenarios and allows the scenarios to be better tailored to the problem at hand. The method is based on a semi-infinite linear programming formulation of the problem that is shown to be solvable with polynomial iteration complexity. A practical column generation method is implemented. The column generation subproblems are polynomial optimization problems; however, they need not be solved to optimality. It is found that the columns in the column generation approach can be efficiently generated by random sampling. The number of scenarios generated matches a lower bound of Tchakaloff's. The rate of convergence of the approximation error is established for continuous integrands, and an improved bound is given for smooth integrands. Extensive numerical experiments are presented in which variants of the proposed method are compared to Monte Carlo and quasi-Monte Carlo methods on both numerical integration problems and stochastic optimization problems. The benefits of being able to match any prescribed set of moments, rather than all moments up to a certain order, is also demonstrated using optimization problems with 100-dimensional random vectors. Here, empirical results show that the proposed approach outperforms Monte Carlo and quasi-Monte Carlo based approaches on the tested problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Understanding performance variability in standard and pipelined parallel Krylov solvers

In this work, we collect data from runs of Krylov subspace methods and pipelined Krylov algorithms in an effort to understand and model the impact of machine noise and other sources of variability on performance. We find large variability of Krylov iterations between compute nodes for standard methods that is reduced in pipelined algorithms, directly supporting conjecture, as well as large variation between statistical distributions of runtimes across iterations. Based on these results, we improve upon a previously introduced nondeterministic performance model by allowing iterations to fluctuate over time. We present our data from runs of various Krylov algorithms across multiple platforms as well as our updated non-stationary model that provides good agreement with observations. We also suggest how it can be used as a predictive tool.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the invariant subspace is large (e.g., over several hundreds or thousands) even though it may still be small relative to the dimension of A. These problems arise from, for example, density functional theory (DFT) based electronic structure calculations for complex materials. The key feature of our algorithm is that it performs fewer Rayleigh–Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency. We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Multistep inversion workflow for 3D long-offset damped elastic waves in the Fourier domain

We present a new workflow for imaging damped three-dimensional elastic wavefields in the Fourier domain. The workflow employs a multiscale imaging approach, in which offset lengths are laddered, where frequency content and damping of the data are changed cyclically. Thus, the inversion process is launched using short-offset and low-frequency data to recover the long spatial wavelength of the image at a shallow depth. Increasing frequency and offset length leads to the recovery of the fine-scale features of the model at greater depths. For the fixed offset, we employ (in the imaging process) a few discrete frequencies with a set of Laplace damping parameters. The forward problem is solved with a finite-difference frequency-domain method based on a massively parallel iterative solver. The inversion code is based upon the solution of a least squares optimisation problem and is solved using a nonlinear gradient method. It is fully parallelised for distributed memory computational platforms. Our full-waveform inversion workflow is applied to the 3D Marmousi-2 and SEG/EAGE Salt models with long-offset data. The maximum inverted frequencies are 6 Hz for the Marmousi model and 2 Hz for the SEG/EAGE Salt model. The detailed structures are imaged successfully up to the depth approximately equal to one-third of the maximum offset length at a resolution consistent with the inverted frequencies.

3D full waveform inversion↗
DOE OSTI2022

Developments and Challenges in the Design of the ITER DRGA

We report the ITER Diagnostic Residual Gas Analyzer (DRGA) will measure the distribution of gas species, i.e., deuterium (D), tritium (T), and impurities, in the divertor exhaust stream and in the plasma periphery, with time resolution relevant to fusion plasma–wall particle dynamics. The uniqueness of the DRGA, over previous implementations of plasma dynamics residual gas analysis, is an integrated approach, combining mass and low-temperature plasma-activated optical spectroscopy, in a differentially pumped analysis station. A further unique feature of the ITER divertor-specific DRGA is an ~8-m separation of the analysis station from the sampled pumping duct, while still maintaining a ~1-s response time for hydrogen isotopic concentrations. ITER DRGA final design activities are strongly benefiting from testing of prototypical DRGA components and methods on present fusion devices, most currently on JET and W7-X. DRGA systems are implemented on both these devices and include sensors (and pumping methods) that are directly relevant to the ITER DRGA design. The recent JET-DTE2 campaign has provided the first experience on operating the combined ITER DRGA sensors with D-T plasmas. While enhancing system design for ITER, this experience on operating devices has also revealed additional engineering challenges, which further guide the continuing final design project. Meanwhile, the recent determination that the ITER DRGA, with slight optimization, will resolve the helium isotopes well enough to support an ITER pre-DT, He-3-based heating scheme, has greatly increased ITER Research Program interest in the DRGA and its implementation well ahead of the DT phase.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

SU-F-J-97: A Joint Registration and Segmentation Approach for Large Bladder Deformations in Adaptive Radiotherapy

Purpose: To improve results of deformable image registration (DIR) in adaptive radiotherapy for large bladder deformations in CT/CBCT pelvis imaging. Methods: A variational multi-modal DIR algorithm is incorporated in a joint iterative scheme, alternating between segmentation based bladder matching and registration. Using an initial DIR to propagate the bladder contour to the CBCT, in a segmentation step the contour is improved by discrete image gradient sampling along all surface normals and adapting the delineation to match the location of each maximum (with a search range of +−5/2mm at the superior/inferior bladder side and step size of 0.5mm). An additional graph-cut based constraint limits the maximum difference between neighboring points. This improved contour is utilized in a subsequent DIR with a surface matching constraint. By calculating an euclidean distance map of the improved contour surface, the new constraint enforces the DIR to map each point of the original contour onto the improved contour. The resulting deformation is then used as a starting guess to compute a deformation update, which can again be used for the next segmentation step. The result is a dense deformation, able to capture much larger bladder deformations. The new method is evaluated on ten CT/CBCT male pelvis datasets, calculating Dice similarity coefficients (DSC) between the final propagated bladder contour and a manually delineated gold standard on the CBCT image. Results: Over all ten cases, an average DSC of 0.93±0.03 is achieved on the bladder. Compared with the initial DIR (0.88±0.05), the DSC is equal (2 cases) or improved (8 cases). Additionally, DSC accuracy of femoral bones (0.94±0.02) was not affected. Conclusion: The new approach shows that using the presented alternating segmentation/registration approach, the results of bladder DIR in the pelvis region can be greatly improved, especially for cases with large variations in bladder volume. Fraunhofer MEVIS received funding from a research grant by Varian Medical Systems.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2013

CT head-scan dosimetry in an anthropomorphic phantom and associated measurement of ACR accreditation-phantom imaging metrics under clinically representative scan conditions

Purpose: To measure radiation absorbed dose and its distribution in an anthropomorphic head phantom under clinically representative scan conditions in three widely used computed tomography (CT) scanners, and to relate those dose values to metrics such as high-contrast resolution, noise, and contrast-to-noise ratio (CNR) in the American College of Radiology CT accreditation phantom.Methods: By inserting optically stimulated luminescence dosimeters (OSLDs) in the head of an anthropomorphic phantom specially developed for CT dosimetry (University of Florida, Gainesville), we measured dose with three commonly used scanners (GE Discovery CT750 HD, Siemens Definition, Philips Brilliance 64) at two different clinical sites (Walter Reed National Military Medical Center, National Institutes of Health). The scanners were set to operate with the same data-acquisition and image-reconstruction protocols as used clinically for typical head scans, respective of the practices of each facility for each scanner. We also analyzed images of the ACR CT accreditation phantom with the corresponding protocols. While the Siemens Definition and the Philips Brilliance protocols utilized only conventional, filtered back-projection (FBP) image-reconstruction methods, the GE Discovery also employed its particular version of an adaptive statistical iterative reconstruction (ASIR) algorithm that can be blended in desired proportions with the FBP algorithm. We did an objective image-metrics analysis evaluating the modulation transfer function (MTF), noise power spectrum (NPS), and CNR for images reconstructed with FBP. For images reconstructed with ASIR, we only analyzed the CNR, since MTF and NPS results are expected to depend on the object for iterative reconstruction algorithms.Results: The OSLD measurements showed that the Siemens Definition and the Philips Brilliance scanners (located at two different clinical facilities) yield average absorbed doses in tissue of 42.6 and 43.1 mGy, respectively. The GE Discovery delivers about the same amount of dose (43.7 mGy) when run under similar operating and image-reconstruction conditions, i.e., without tube current modulation and ASIR. The image-metrics analysis likewise showed that the MTF, NPS, and CNR associated with the reconstructed images are mutually comparable when the three scanners are run with similar settings, and differences can be attributed to different edge-enhancement properties of the applied reconstruction filters. Moreover, when the GE scanner was operated with the facility's scanner settings for routine head exams, which apply 50% ASIR and use only approximately half of the 100%-FBP dose, the CNR of the images showed no significant change. Even though the CNR alone is not sufficient to characterize the image quality and justify any dose reduction claims, it can be useful as a constancy test metric.Conclusions: This work presents a straightforward method to connect direct measurements of CT dose with objective image metrics such as high-contrast resolution, noise, and CNR. It demonstrates that OSLD measurements in an anthropomorphic head phantom allow a realistic and locally precise estimation of magnitude and spatial distribution of dose in tissue delivered during a typical CT head scan. Additional objective analysis of the images of the ACR accreditation phantom can be used to relate the measured doses to high contrast resolution, noise, and CNR.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2016

Effect of reconstruction methods and x-ray tube current–time product on nodule detection in an anthropomorphic thorax phantom: A crossed-modality JAFROC observer study

Purpose: To evaluate nodule detection in an anthropomorphic chest phantom in computed tomography (CT) images reconstructed with adaptive iterative dose reduction 3D (AIDR{sup 3D}) and filtered back projection (FBP) over a range of tube current–time product (mAs). Methods: Two phantoms were used in this study: (i) an anthropomorphic chest phantom was loaded with spherical simulated nodules of 5, 8, 10, and 12 mm in diameter and +100, −630, and −800 Hounsfield units electron density; this would generate CT images for the observer study; (ii) a whole-body dosimetry verification phantom was used to ultimately estimate effective dose and risk according to the model of the BEIR VII committee. Both phantoms were scanned over a mAs range (10, 20, 30, and 40), while all other acquisition parameters remained constant. Images were reconstructed with both AIDR{sup 3D} and FBP. For the observer study, 34 normal cases (no nodules) and 34 abnormal cases (containing 1–3 nodules, mean 1.35 ± 0.54) were chosen. Eleven observers evaluated images from all mAs and reconstruction methods under the free-response paradigm. A crossed-modality jackknife alternative free-response operating characteristic (JAFROC) analysis method was developed for data analysis, averaging data over the two factors influencing nodule detection in this study: mAs and image reconstruction (AIDR{sup 3D} or FBP). A Bonferroni correction was applied and the threshold for declaring significance was set at 0.025 to maintain the overall probability of Type I error at α = 0.05. Contrast-to-noise (CNR) was also measured for all nodules and evaluated by a linear least squares analysis. Results: For random-reader fixed-case crossed-modality JAFROC analysis, there was no significant difference in nodule detection between AIDR{sup 3D} and FBP when data were averaged over mAs [F(1, 10) = 0.08, p = 0.789]. However, when data were averaged over reconstruction methods, a significant difference was seen between multiple pairs of mAs settings [F(3, 30) = 15.96, p < 0.001]. Measurements of effective dose and effective risk showed the expected linear dependence on mAs. Nodule CNR was statistically higher for simulated nodules on images reconstructed with AIDR{sup 3D} (p < 0.001). Conclusions: No significant difference in nodule detection performance was demonstrated between images reconstructed with FBP and AIDR{sup 3D}. mAs was found to influence nodule detection, though further work is required for dose optimization.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

A multiscale flux basis for mortar mixed discretizations of Stokes–Darcy flows

A multiscale flux basis algorithm is developed for the Stokes–Darcy flow problem. The method is based on a non-overlapping domain decomposition algorithm, where the global problem is reduced to a coarse scale mortar interface problem that is solved by an iterative solver. Subdomain solves are required at each interface iteration, so the cost for the method without a multiscale basis can be high when the number of subdomains or the condition number of the interface problem is large. Here the proposed algorithm involves precomputing a multiscale flux basis, which consists of the flux (or velocity trace) response from each mortar degree of freedom. It is computed by each subdomain independently before the interface iteration begins. The subdomain solves required at each iteration are substituted by a linear combination of the multiscale basis. This may lead to a significant reduction in computational cost since the number of subdomain solves is fixed, depending only on the number of mortar degrees of freedom associated with a subdomain. Several numerical examples are carried out to demonstrate the efficiency of the multiscale flux basis implementation for large-scale Stokes–Darcy problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Deep Reinforcement Learning Based Volt-VAR Optimization in Smart Distribution Systems

This paper develops a model-free volt-VAR optimization (VVO) algorithm via multi-agent deep reinforcement learning (DRL) in unbalanced distribution systems. This method is novel since we cast the VVO problem in distribution networks to an intelligent deep Q-network (DQN) framework, which avoids solving a specific optimization model directly when facing time-varying operating conditions in the systems. We consider statuses/ratios of switchable capacitors, voltage regulators, and smart inverters installed at distributed generators as the action variables of the agents. A delicately designed reward function guides these agents to interact with the distribution system, in the direction of reinforcing voltage regulation and power loss reduction simultaneously. The forward-backward sweep method for radial three-phase distribution systems provides accurate power flow results within a few iterations to the DRL environment. The proposed method realizes the dual goals for VVO. We test this algorithm on the unbalanced IEEE 13-bus and 123-bus systems. Numerical simulations validate the excellent performance of this method in voltage regulation and power loss reduction.

41 EE - Solar Energy Technologies Office (EE-4S)↗
DOE OSTI2024

SLICE

For most of their lifetime, pebble-bed reactors (PBRs) operate at an equilibrium state in which the core is filled with fuel pebbles at various levels of burnup. A fuel pebble travels multiple times in so-called passes through the reactor before it reaches its target discharge burnup and is replaced with a fresh fuel pebble. Given the stochastic nature of the fuel pebble travel paths and consequently the individual fuel pebble histories, it is not possible with standard methods developed for traditional reactor concepts to calculate the fuel inventory in the reactor core. An iterative approach, the SCALE Leap-In method for Cores at Equilibrium (SLICE), was developed to generate region-average fuel inventory for a PBR. The SLICE code enables automatic generation of input files for the SCALE code system (https://www.ornl.gov/scale), management of the SCALE result files, and analysis of results.

Bostelmann, Friederike [Oak Ridge National Laborat↗
DOE OSTI2016

3D exemplar-based random walks for tooth segmentation from cone-beam computed tomography images

Purpose: Tooth segmentation is an essential step in acquiring patient-specific dental geometries from cone-beam computed tomography (CBCT) images. Tooth segmentation from CBCT images is still a challenging task considering the comparatively low image quality caused by the limited radiation dose, as well as structural ambiguities from intercuspation and nearby alveolar bones. The goal of this paper is to present and discuss the latest accomplishments in semisupervised tooth segmentation with adaptive 3D shape constraints. Methods: The authors propose a 3D exemplar-based random walk method of tooth segmentation from CBCT images. The proposed method integrates semisupervised label propagation and regularization by 3D exemplar registration. To begin with, the pure random walk method is to get an initial segmentation of the teeth, which tends to be erroneous because of the structural ambiguity of CBCT images. And then, as an iterative refinement, the authors conduct a regularization by using 3D exemplar registration, as well as label propagation by random walks with soft constraints, to improve the tooth segmentation. In the first stage of the iteration, 3D exemplars with well-defined topologies are adapted to fit the tooth contours, which are obtained from the random walks based segmentation. The soft constraints on voxel labeling are defined by shape-based foreground dentine probability acquired by the exemplar registration, as well as the appearance-based probability from a support vector machine (SVM) classifier. In the second stage, the labels of the volume-of-interest (VOI) are updated by the random walks with soft constraints. The two stages are optimized iteratively. Instead of the one-shot label propagation in the VOI, an iterative refinement process can achieve a reliable tooth segmentation by virtue of exemplar-based random walks with adaptive soft constraints. Results: The proposed method was applied for tooth segmentation of twenty clinically captured CBCT images. Three metrics, including the Dice similarity coefficient (DSC), the Jaccard similarity coefficient (JSC), and the mean surface deviation (MSD), were used to quantitatively analyze the segmentation of anterior teeth including incisors and canines, premolars, and molars. The segmentation of the anterior teeth achieved a DSC up to 98%, a JSC of 97%, and an MSD of 0.11 mm compared with manual segmentation. For the premolars, the average values of DSC, JSC, and MSD were 98%, 96%, and 0.12 mm, respectively. The proposed method yielded a DSC of 95%, a JSC of 89%, and an MSD of 0.26 mm for molars. Aside from the interactive definition of label priors by the user, automatic tooth segmentation can be achieved in an average of 1.18 min. Conclusions: The proposed technique enables an efficient and reliable tooth segmentation from CBCT images. This study makes it clinically practical to segment teeth from CBCT images, thus facilitating pre- and interoperative uses of dental morphologies in maxillofacial and orthodontic treatments.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2023

Feasibility of raised inner strike point equilibria scenario in ITER for detritiation from beryllium co-deposits

Abstract In ITER, tritium retention primarily occurs through co-deposition with beryllium. To avoid exceeding the strict tritium inventory limit, efficient tritium recovery techniques are essential. Baking is the ITER baseline for tritium recovery, but its effectiveness in removing tritium from thick beryllium layers is limited. A raised strike point scenario is considered an alternative method for removing tritium from the ITER inner vertical divertor target by heating components via plasma flux. This paper presents SOLPS-ITER code simulations conducted under various conditions, assessing the divertor performance and tritium outgassing of the raised strike point scenario. As the strike point is raised, recycled neutrals are not efficiently baffled by the dome and scrape-off layer, significantly changing the neutral trajectory and ionization source distribution. This improves detachment accessibility but worsens core-edge compatibility compared to the baseline scenario. However, in the partially detached condition, the impact of raising the strike point, perpendicular transport, and q 95 on target heat flux is not significant, as it primarily scales with the input power. Target heat flux is translated to target surface temperature using a simplified heat transfer model that considers the 3D target monoblock geometry and active cooling condition, excluding Be layer thermal properties. For partially detached divertor conditions, the bulk tungsten monoblock surface temperature remains below the baking temperature, which is insufficient for efficient tritium outgassing under the actively cooled ITER divertor condition. However, considering the potential thermal contact resistance between the beryllium and tungsten layers, which may significantly impact temperature distribution, the temperature of the beryllium layer can be raised to a level sufficient for efficient tritium outgassing. Therefore, the raised strike point scenario can be considered as an alternative in-vessel tritium removal technique.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2013

An angularly refineable phase space finite element method with approximate sweeping procedure

An angularly refineable phase space finite element method is proposed to solve the neutron transport equation. The method combines the advantages of two recently published schemes. The angular domain is discretized into small patches and patch-wise discontinuous angular basis functions are restricted to these patches, i.e. there is no overlap between basis functions corresponding to different patches. This approach yields block diagonal Jacobians with small block size and retains the possibility for S{sub n}-like approximate sweeping of the spatially discontinuous elements in order to provide efficient preconditioners for the solution procedure. On the other hand, the preservation of the full FEM framework (as opposed to collocation into a high-order S{sub n} scheme) retains the possibility of the Galerkin interpolated connection between phase space elements at arbitrary levels of discretization. Since the basis vectors are not orthonormal, a generalization of the Riemann procedure is introduced to separate the incoming and outgoing contributions in case of unstructured meshes. However, due to the properties of the angular discretization, the Riemann procedure can be avoided at a large fraction of the faces and this fraction rapidly increases as the level of refinement increases, contributing to the computational efficiency. In this paper the properties of the discretization scheme are studied with uniform refinement using an iterative solver based on the S{sub 2} sweep order of the spatial elements. The fourth order convergence of the scalar flux is shown as anticipated from earlier schemes and the rapidly decreasing fraction of required Riemann faces is illustrated. (authors)

97 MATHEMATICS AND COMPUTING↗