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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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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Kinetics and Thermodynamics of Sr Permeation in CeO 2 -Based Barrier Layers for Solid-Oxide Electrolyzer Cells

Solid-oxide electrolyzer cells (SOECs) convert steam to hydrogen efficiently at high temperatures. However, during operation, the diffusion of cations or impurities through the cells due to electrode degradation can cause unwanted secondary phases to form, which may degrade device performance. Here, in this study, we use atomistic and mesoscale simulations coupled with experimental analysis to study the diffusion of Sr through the Gd-doped CeO 2 (GDC) barrier layer used to protect the yttria-stabilized zirconia (YSZ) electrolyte in SOECs. From our atomistic calculations, we find Sr diffusion to be negligibly slow in bulk GDC; however, surface diffusion is much more favorable. Subsequent mesoscale simulations show that Sr diffusion is activated when the porosity of GDC exceeds ∼10% and significantly exceeds diffusion in bulk and grain boundary regions. We also find that SrO-based species can accumulate at GDC surfaces; however, SrO aggregation and coarsening will be limited by the large lattice mismatch between GDC and SrO. Energy-dispersive X-ray spectroscopy (EDS) and electron diffraction confirm that Sr can accumulate within GDC pores and form disperse Sr-containing secondary phases. Altogether, Sr diffusion in dense GDC is unlikely to give rise to thick SrO layers, which would severely limit device performance. The formation of Sr-containing secondary phases can largely be avoided by restricting the porosity of the GDC layer as much as possible.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Europium diffusion in IG-110 nuclear graphite

Europium diffusion in graphite has been recognized to be of interest in high-temperature gas reactor safety analysis, particularly as an indicator of strontium diffusion. However, no measurements of europium diffusion coefficients have been reported in the literature. In this work, the effective diffusion coefficient of europium was measured using a time-release method. Natural europium was loaded into pre-milled unirradiated IG-110 graphite spheres using a pressurized acid digestion vessel. The time-release experiments were performed in the temperature range 1823 K – 1973 K using a SiC diffusion cell connected to an inductively-coupled plasma mass spectrometer (ICP-MS) via a He gas line. Finally, the results of this work are: D Eu, IG-110 = (1.5 x 10 -3 m 2 /s) exp $(\frac{-2.87\mathrm{x}10^{5} J/mol}{RT})$ This effective diffusion coefficient can be used to aid in predictive modelling of europium transport in HTGRs.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Steric effects of central dogma processes on the compaction and segregation of bacterial nucleoids

The bacterial cytoplasm is characterized by a distinctive membrane-less organelle, the nucleoid, which harbors the chromosomal DNA. Here, we investigate the steric effects of dynamic processes associated with transcription and translation on the structure of this organelle using coarse-grained molecular dynamics simulations that incorporate out-of-equilibrium reactions. Our model captures the scale of the entire cell and incorporates a reaction-diffusion system for ribosomes and polyribosomes, coupling their nonequilibrium kinetics to DNA excluded volume interactions. Our findings demonstrate that out-of-equilibrium reactions increase the size of the nucleoid and the number of ribosomes in this subcellular region. In addition, we show that nucleoid size is proportional to transcriptional activity. Our model reproduces the time-dependent change in nucleoid size observed in rifampicin treatment experiments, where the pool of polyribosomes is depleted. Furthermore, we find that these active processes are essential for complete sister chromosome separation and correct nucleoid positioning within the cell. Overall, our study reveals the effects of the central dogma processes on the internal organization and localization of bacterial nucleoids.

Chang, Mu-Hung [Univ. of Tennessee, Knoxville, TN ↗
DOE OSTI2025

Transient fuel performance analysis for the preliminary fuel concept of general atomics fast modular reactor

This study investigates the transient fuel performance of General Atomics Fast Modular Reactor (GA-FMR) during accident scenarios, focusing on the behavior of its innovative fuel system that combines high-assay low enriched uranium dioxide (HALEUO2) fuel with SiGA® ceramic matrix composite silicon carbide cladding. The preliminary fuel design’s response was analyzed during reactivity-initiated accidents (RIA) and loss of coolant accidents (LOCA) using BISON fuel performance analysis code, which included both the diffusion enhanced and BISON-FASTGRASS coupled UO 2 models. The RIA analysis demonstrated that effective reactivity control reduced fuel temperature, though with transient fission gas release resulting in additional tensile stress state on the cladding. LOCA simulations revealed differing predictions between the two models: the BISON UO 2 model showed more transient fission gas release but minimal pellet expansion, while the BISON-FASTGRASS UO 2 model predicted less pronounced fission gas release but more fuel swelling and thermal expansion, potentially leading to pellet-cladding mechanical interaction. Here, these findings highlight critical areas for fuel design optimization and identify knowledge gaps requiring further experimental and computational investigation to advance GA-FMR fuel development.

Lee, Soon K. [Argonne National Laboratory (ANL), A↗
DOE OSTI2023

Direct Conversion of Methane to Ethylene and Acetylene over an Iron-Based Metal–Organic Framework

Conversion of methane (CH 4 ) to ethylene (C 2 H 4 ) and/or acetylene (C 2 H 2 ) enables routes to a wide range of products directly from natural gas. However, high reaction temperatures and pressures are often required to activate and convert CH 4 controllably, and separating C 2+ products from unreacted CH 4 can be challenging. Here, we report the direct conversion of CH 4 to C 2 H 4 and C 2 H 2 driven by non-thermal plasma under ambient (25 °C and 1 atm) and flow conditions over a metal–organic framework material, MFM-300(Fe). The selectivity for the formation of C 2 H 4 and C 2 H 2 reaches 96% with a high time yield of 334 μmol g cat –1 h –1 . At a conversion of 10%, the selectivity to C 2+ hydrocarbons and time yield exceed 98% and 2056 μmol g cat –1 h –1 , respectively, representing a new benchmark for conversion of CH 4 . In situ neutron powder diffraction, inelastic neutron scattering and solid-state nuclear magnetic resonance, electron paramagnetic resonance (EPR), and diffuse reflectance infrared Fourier transform spectroscopies, coupled with modeling studies, reveal the crucial role of Fe–O(H)–Fe sites in activating CH 4 and stabilizing reaction intermediates via the formation of an Fe–O(CH 3 )–Fe adduct. In addition, a cascade fixed-bed system has been developed to achieve online separation of C 2 H 4 and C 2 H 2 from unreacted CH 4 for direct use. Integrating the processes of CH 4 activation, conversion, and product separation within one system opens a new avenue for natural gas utility, bridging the gap between fundamental studies and practical applications in this area.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

A 3D nm-thin biomimetic membrane for ultimate molecular separation

Multi-functional membranes with high permeance and selectivity that can mimic nature's designs have tremendous industrial and bio-medical applications. In this paper, we report a novel concept of a 3D nanometer (nm)-thin membrane that can overcome the shortcomings of conventional membrane structures. Our 3D membrane is composed of two three-dimensionally interwoven channels that are separated by a continuous nm-thin amorphous TiO 2 layer. This 3D architecture dramatically increases the surface area by 6000 times, coupled with an ultra-short diffusion distance through the 2–4 nm-thin selective layer that allows for ultrafast gas and water transport, ~900 l m –2 h –1 bar –1 . The 3D membrane also exhibits a very high ion rejection (R ~ 100% for potassium ferricyanide) due to the combined size- and charge-based exclusion mechanisms. The combination of high ion rejection and ultrafast permeation makes our 3DM superior to the state-of-the-art high-flux membranes whose performances are limited by the flux-rejection tradeoff. Furthermore, its ultimate Li+ selectivity over polysulfide or gas can potentially solve major technical challenges in energy storage applications, such as lithium–sulfur or lithium–O 2 batteries.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Real time quantification of mixed ion and electron transfer associated with the doping of poly(3-hexylthiophene)

Conjugated polymers have promising applications in electronics and energy storage due to the polymer's tunable conductivity and redox activity. For example, the conductivity of poly(3-hexylthiophene) (P3HT) is heavily dependent upon the doping level and the dopant type. This feature becomes especially important when considering P3HT or similar conjugated polymers for devices that require switching between electronic states (conductive vs. insulating). In this study, the mechanism of mixed ion-electron transfer studied using electrochemical quartz crystal microbalance with dissipation monitoring (EQCM-D) is discussed. During cyclic voltammetry and galvanostatic charge–discharge experiments, the mass change of a P3HT film is monitored in real time. Distinct mass transfer regions are quantified as a function of doping level and potential, which are then correlated to changes with in situ conductance and spectroelectrochemical response. To identify the time scale at which the doping reaction transitions from kinetic to diffusion control, electrochemical impedance spectroscopy is coupled with EQCM-D. Finally, this work gives valuable insight into the nature of mixed ion–electron transfer, including its time scale, as it relates to the electronic properties of P3HT.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Learning how structures form in drift-wave turbulence

Drift-wave turbulence produces anomalous transport via cross-correlations between fluctuations. This transport has profound implications for confinement, structure formation, and virtually all aspects of the non-linear turbulent dynamics. Herein, we use a data-driven method based on deep learning in order to study turbulent transport in the 2D Hasegawa–Wakatani system and infer a reduced mean-field model from numerical solution. In addition to the usual turbulent diffusion, we find an effect which couples the particle flux to the local gradient of vorticity, which tends to modulate the density profile. The direct coupling to the shear is relatively weak. In addition, the deep learning method finds a model for spontaneous zonal flow generation by negative viscosity, stabilized by non-linear and hyperviscous terms. We compare these results to analytic calculations using quasilinear theory and wave kinetics, finding qualitative agreement, though the calculations miss certain higher-order effects. A simplified, 1-D model for the evolution of the profile, flow, and intensity based on the deep learning results is solved numerically and compared to previous models for staircasing based on bistability. We see that the physics uncovered by the deep learning method provided simple explanations for the formation of zonal structures in the density, flow, and turbulence fields. We highlight the important role of symmetry in the deep learning method and speculate on the portability of the method to other applications.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Space-Time Block Preconditioning for Incompressible Flow

Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional message passing interface processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time to solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Finally, the results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iteration count when applied to a nonlinear Navier--Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Multifidelity computing for coupling full and reduced order models

Hybrid physics-machine learning models are increasingly being used in simulations of transport processes. Many complex multiphysics systems relevant to scientific and engineering applications include multiple spatiotemporal scales and comprise a multifidelity problem sharing an interface between various formulations or heterogeneous computational entities. To this end, we present a robust hybrid analysis and modeling approach combining a physics-based full order model (FOM) and a data-driven reduced order model (ROM) to form the building blocks of an integrated approach among mixed fidelity descriptions toward predictive digital twin technologies. At the interface, we introduce a long short-term memory network to bridge these high and low-fidelity models in various forms of interfacial error correction or prolongation. The proposed interface learning approaches are tested as a new way to address ROM-FOM coupling problems solving nonlinear advection-diffusion flow situations with a bifidelity setup that captures the essence of a broad class of transport processes.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2021

Performance Improvements for the Griffin Transport Solvers

Griffin is a Multiphysics Object-Oriented Simulation Environment based reactor multiphysics analysis application jointly developed by Idaho National Laboratory and Argonne National Laboratory. Griffin includes a variety of deterministic radiation transport solvers for fixed source, k-eigenvalue, adjoint, and subcritical multiplication, as well as transient solvers for point-kinetics, improved quasi-static, and spatial dynamics. A code assessment performed in FY-20 identified two significant issues with the transport solvers in Griffin: first, the primary heterogeneous SN (discrete ordinates) transport solver based on continuous finite element methods required significant mesh refinement and higher memory usage compared to solvers based on the method of characteristic for equivalent accuracy. Second, the homogeneous PN (spherical harmonics expansion) transport solver did not adequately support polynomial refinement, which is a feature usually required for problems with spatial homogenization and pronounced streaming, typical in fast or gas-cooled reactor systems. To address the first issue, the development effort focused on the more promising discontinuous finite element method (DFEM)-based SN transport solver in Griffin. The addition of an asynchronous parallel transport sweeper and coarse mesh finite difference (CMFD) acceleration have rendered a superior heterogeneous SN transport capability for multiphysics problems that requires far less computing resources in terms of both CPU time and memory usage. This is demonstrated with typical thermal- and fast-spectrum reactor benchmark problems, including 2D Transient Reactor Test, 3D Advanced Burner Test Reactor (ABTR), and 2D and 3D Empire microreactor. For the second issue, the development effort focused on a new transport solver based on the hybrid finite element PN method (HFEM-PN), equivalent to the variational nodal method, as well as a new diffusion solver based on HFEM-Diffusion. This solver is intended for homogenized domains with multiphysics coupling (i.e., supports mesh displacement, seamless temperature feedback, etc.). Initial calculations with the HFEM-Diffusion implementation show very good parallel efficiency for the residual evaluations with the 2D ABTR benchmark. A future development effort will be centered on further improvements to the CMFD, HFEM-PN, and DFEM diffusion solvers to ensure Griffin meets performance and software quality assurance requirements for advanced reactor design and analysis.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2021

Understanding Selectivity in CO2 Hydrogenation to Methanol for MoP Nanoparticle Catalysts Using In Situ Techniques

Molybdenum phosphide (MoP) catalyzes the hydrogenation of CO, CO2, and their mixtures to methanol, and it is investigated as a high-activity catalyst that overcomes deactivation issues (e.g., formate poisoning) faced by conventional transition metal catalysts. MoP as a new catalyst for hydrogenating CO2 to methanol is particularly appealing for the use of CO2 as chemical feedstock. Herein, we use a colloidal synthesis technique that connects the presence of MoP to the formation of methanol from CO2, regardless of the support being used. By conducting a systematic support study, we see that zirconia (ZrO2) has the striking ability to shift the selectivity towards methanol by increasing the rate of methanol conversion by two orders of magnitude compared to other supports, at a CO2 conversion of 1.4% and methanol selectivity of 55.4%. In situ X-ray Absorption Spectroscopy (XAS) and in situ X-ray Diffraction (XRD) indicate that under reaction conditions the catalyst is pure MoP in a partially crystalline phase. Results from Diffuse Reflectance Infrared Fourier Transform Spectroscopy coupled with Temperature Programmed Surface Reaction (DRIFTS-TPSR) point towards a highly reactive monodentate formate intermediate stabilized by the strong interaction of MoP and ZrO2. This study definitively shows that the presence of a MoP phase leads to methanol formation from CO2, regardless of support and that the formate intermediate on MoP governs methanol formation rate.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Constraints on Metastable Helium in the Atmospheres of WASP-69b and WASP-52b with Ultranarrowband Photometry

Infrared observations of metastable 2{sup 3}S helium absorption with ground- and space-based spectroscopy are rapidly maturing, as this species is a unique probe of exoplanet atmospheres. Specifically, the transit depth in the triplet feature (with vacuum wavelengths near 1083.3 nm) can be used to constrain the temperature and mass-loss rate of an exoplanet’s upper atmosphere. Here, we present a new photometric technique to measure metastable 2{sup 3}S helium absorption using an ultranarrowband filter (FWHM 0.635 nm) coupled to a beam-shaping diffuser installed in the Wide-field Infrared Camera on the 200 inch Hale Telescope at Palomar Observatory. We use telluric OH lines and a helium arc lamp to characterize refractive effects through the filter and to confirm our understanding of the filter transmission profile. We benchmark our new technique by observing a transit of WASP-69b and detect an excess absorption of 0.498% ± 0.045% (11.1σ), consistent with previous measurements after considering our bandpass. We then use this method to study the inflated gas giant WASP-52b and place a 95th percentile upper limit on excess absorption in our helium bandpass of 0.47%. Using an atmospheric escape model, we constrain the mass-loss rate for WASP-69b to be 5.25{sub −0.46}{sup +0.65}×10{sup −4} M{sub J} Gyr{sup −1} (3.32{sub −0.56}{sup +0.67}×10{sup −3} M{sub J} Gyr{sup −1}) at 7000 K (12,000 K). Additionally, we set an upper limit on the mass-loss rate of WASP-52b at these temperatures of 2.1×10{sup −4} M{sub J} Gyr{sup −1} (2.1×10{sup −3} M{sub J} Gyr{sup −1}). These results show that ultranarrowband photometry can reliably quantify absorption in the metastable helium feature.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Performance of Diffusion Monte Carlo Calculations for Predicting the Relative Energies of Quinoidal and Nonquinoidal Species

Coupled cluster singles and doubles with perturbative triples [CCSD(T)] and single determinant fixed-node diffusion Monte Carlo (SD-DMC) have emerged as two of the most useful methods for providing benchmark reaction and interaction energies of chemical systems without strong static correlation. The errors in DMC energies are dominated by an inexact description of the nodal surfaces for electron exchange. One of the main approaches to addressing the fixed-node error is to use multideterminant (MD) trial wave functions. We consider here the energy differences between pairs of related molecules with aromatic and quinoidal structures as well as between quinoidal isomers. Quinoidal systems tend to have some diradical character, leading one to anticipate that SD-DMC calculations may face challenges in accurately describing their energetics. The MD trial wave functions were generated from the complete active space calculations. A comparison is made with the predictions of well-converged CCSD(T) calculations.

basis sets↗
DOE OSTI2021

Zinc binding alters the conformational dynamics and drives the transport cycle of the cation diffusion facilitator YiiP

YiiP is a secondary transporter that couples Zn 2+ transport to the proton motive force. Structural studies of YiiP from prokaryotes and Znt8 from humans have revealed three different Zn 2+ sites and a conserved homodimeric architecture. These structures define the inward-facing and outward-facing states that characterize the archetypal alternating access mechanism of transport. To study the effects of Zn 2+ binding on the conformational transition, we use cryo-EM together with molecular dynamics simulation to compare structures of YiiP from Shewanella oneidensis in the presence and absence of Zn 2+ . To enable single-particle cryo-EM, we used a phage-display library to develop a Fab antibody fragment with high affinity for YiiP, thus producing a YiiP/Fab complex. To perform MD simulations, we developed a nonbonded dummy model for Zn 2+ and validated its performance with known Zn 2+ -binding proteins. Using these tools, we find that, in the presence of Zn 2+ , YiiP adopts an inward-facing conformation consistent with that previously seen in tubular crystals. After removal of Zn 2+ with high-affinity chelators, YiiP exhibits enhanced flexibility and adopts a novel conformation that appears to be intermediate between inward-facing and outward-facing states. This conformation involves closure of a hydrophobic gate that has been postulated to control access to the primary transport site. Comparison of several independent cryo-EM maps suggests that the transition from the inward-facing state is controlled by occupancy of a secondary Zn 2+ site at the cytoplasmic membrane interface. This work enhances our understanding of individual Zn 2+ binding sites and their role in the conformational dynamics that govern the transport cycle.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2024

Ion diffusion retarded by diverging chemical susceptibility

Abstract For first-order phase transitions, the second derivatives of Gibbs free energy (specific heat and compressibility) diverge at the transition point, resulting in an effect known as super-elasticity along the pressure axis, or super-thermicity along the temperature axis. Here we report a chemical analogy of these singularity effects along the atomic doping axis, where the second derivative of Gibbs free energy (chemical susceptibility) diverges at the transition point, leading to an anomalously high energy barrier for dopant diffusion in co-existing phases, an effect we coin as super-susceptibility. The effect is realized in hydrogen diffusion in vanadium dioxide (VO 2 ) with a metal-insulator transition (MIT). We show that hydrogen faces three times higher energy barrier and over one order of magnitude lower diffusivity when it diffuses across a metal-insulator domain wall in VO 2 . The additional energy barrier is attributed to a volumetric energy penalty that the diffusers need to pay for the reduction of latent heat. The super-susceptibility and resultant retarded atomic diffusion are expected to exist universally in all phase transformations where the transformation temperature is coupled to chemical composition, and inspires new ways to engineer dopant diffusion in phase-coexisting material systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Numerical Study of PAHs and Soot Emissions from Gasoline–Methanol, Gasoline–Ethanol, and Gasoline– n -Butanol Blend Surrogates

Soot formation is an intricate phenomenon, and soot propensity of a fuel is interwoven with the fuel composition, physical and chemical properties, and combustion environment. Here, the present study examines the hypothesis that in addition to the chemical composition of the fuel, the sooting nature of the fuel is closely coupled with its chemical property known as octane sensitivity (S). With this motivation, the present study numerically investigates the effects of gasoline surrogate composition and its property, octane sensitivity (S), on polycyclic aromatic hydrocarbons (PAHs) and soot emissions. Four-component toluene primary reference fuel (TPRF)–alcohol blends, comprising iso-octane, n-heptane, toluene, and one of the three different alcohols- methanol, ethanol, and n-butanol, are used as gasoline surrogates. A total of 320 TPRF–alcohol mixtures, with S in the range of 1–10, are examined under laminar counterflow diffusion flame conditions. A detailed chemical mechanism coupled with a comprehensive soot model, which includes reactions for soot inception, surface growth, PAH condensation, and oxidation, is adopted. The analysis indicates that the toluene content in the fuel mixture has a prominent effect, while the alcohol content and octane sensitivity of the fuel have a weak correlation with the PAHs and soot. Thus, it is not clear if any of these three variables, namely, toluene content in the fuel, alcohol content in the fuel, and S, are individually sufficient to characterize the PAHs and soot across various blends. For this reason, a new variable (X CHO ) based on the elemental composition of the fuel mixture is identified and it is shown that X CHO along with S of the fuel characterize soot emissions satisfactorily. Further, a reaction path analysis indicates that the efficacy of alcohols in reducing soot emissions follows the order: methanol > ethanol > n-butanol.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Efficient smoothed particle radiation hydrodynamics I: Thermal radiative transfer

This work presents efficient solution techniques for radiative transfer in the smoothed particle hydrodynamics discretization. Two choices that impact efficiency are how the material and radiation energy are coupled, which determines the number of iterations needed to converge the emission source, and how the radiation diffusion equation is solved, which must be done in each iteration. The coupled material and radiation energy equations are solved using an inexact Newton iteration scheme based on nonlinear elimination, which reduces the number of Newton iterations needed to converge within each time step. During each Newton iteration, the radiation diffusion equation is solved using Krylov iterative methods with a multigrid preconditioner, which abstracts and optimizes much of the communication when running in parallel. The code is verified for an infinite medium problem, a one-dimensional Marshak wave, and a two and three-dimensional manufactured problem, and exhibits first-order convergence in time and second-order convergence in space. For these problems, the number of iterations needed to converge the inexact Newton scheme and the diffusion equation is independent of the number of spatial points and the number of processors.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗