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DOE OSTI2025

Understanding the structure and mechanism of Na + diffusion in NASICON solid-state electrolytes and the effect of Sc- and Al/Y-substitution

NASICON (sodium superionic conductor) based ceramics are one of the most promising classes of solid-state electrolytes for all-solid-state batteries. However, the mechanism of sodium ion diffusion is not understood in great detail since there is still a discrepancy between reported average structure models, local structures, and the number and position of sodium sites. To close this gap, we investigate the underlying diffusion mechanism and structural changes governing the Na + transport in Na 3.4 Zr 2 Si 2.4 P 0.6 O 12 using quasielastic neutron scattering (QENS) and powder X-ray diffraction (XRD). In the temperature range from 298 K to 640 K, the correlations between structural changes of a monoclinic C2/c to rhombohedral R $\bar{3}$c phase transition and the result of ion diffusion are investigated. The analysis of the quasielastic neutron scattering data reveals two quasielastic components corresponding to the Chudley-Elliott jump-diffusion model. It clearly shows two different Na + diffusion processes, local and long-range, on two different time and length scales and allows calculations of their corresponding activation energies. Additionally, the effects of Sc 3+ and Al 3+ /Y 3+ aliovalent substitution of Zr 4+ ions on the crystal structure and Na + diffusion are also studied. We can distinguish a local, chain, and cross-chain diffusion mechanism based on correlated QENS and XRD comparison of relevant nearest crystallographic Na–Na distances. The results reveal that the Na + diffusion in these NASICONs is three-dimensional and can provide guidelines on how dopants and changes in the crystal structure can affect the Na + conductivity.

25 ENERGY STORAGE↗

DOE OSTI2015

Determination of the effective diffusivity of water in a poly (methyl methacrylate) membrane containing carbon nanotubes using kinetic Monte Carlo simulations

A kinetic Monte Carlo (kMC) simulation algorithm is developed for computing the effective diffusivity of water molecules in a poly(methyl methacrylate) (PMMA) matrix containing carbon nanotubes (CNTs) at several loadings. The simulations are conducted on a cubic lattice to the bonds of which rate constants are assigned governing the elementary jump events of water molecules from one lattice site to another. Lattice sites belonging to PMMA domains of the membrane are assigned different rates than lattice sites belonging to CNT domains. Values of these two rate constants are extracted from available numerical data for water diffusivity within a PMMA matrix and a CNT pre-computed on the basis of independent atomistic molecular dynamics simulations, which show that water diffusivity in CNTs is 3 orders of magnitude faster than in PMMA. Our discrete-space, continuum-time kMC simulation results for several PMMA-CNT nanocomposite membranes (characterized by different values of CNT length L and diameter D and by different loadings of the matrix in CNTs) demonstrate that the overall or effective diffusivity, D{sub eff}, of water in the entire polymeric membrane is of the same order of magnitude as its diffusivity in PMMA domains and increases only linearly with the concentration C (vol. %) in nanotubes. For a constant value of the concentration C, D{sub eff} is found to vary practically linearly also with the CNT aspect ratio L/D. The kMC data allow us to propose a simple bilinear expression for D{sub eff} as a function of C and L/D that can describe the numerical data for water mobility in the membrane extremely accurately. Additional simulations with two different CNT configurations (completely random versus aligned) show that CNT orientation in the polymeric matrix has only a minor effect on D{sub eff} (as long as CNTs do not fully penetrate the membrane). We have also extensively analyzed and quantified sublinear (anomalous) diffusive phenomena over small to moderate times and correlated them with the time needed for penetrant water molecules to explore the available large, fast-diffusing CNT pores before Fickian diffusion is reached.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Defect-mediated diffusion of implanted Mg in GaN: Suppressing dopant redistribution by sequential thermal and microwave annealing

The diffusion behavior of Mg in Mg/N co-implanted GaN is investigated in response to a set of annealing conditions and methodologies, namely, 1000 °C/30 min thermal anneal, by high-temperature pulsed gyrotron microwave annealing at 1420 or 1500 °C, or by thermal and microwave annealing, sequentially. After 1000 °C annealing, the diffusion of Mg in GaN is found to be negligible, as measured by secondary ion mass spectrometry. Annealing by gyrotron microwave annealing alone induces the diffusion of Mg at a rate on the order of 10 −12 cm 2 /s. However, the use of a thermal anneal before microwave gyrotron annealing reduces this rate by an order of magnitude to 10 −13 cm 2 /s. We find that a model that considers Mg diffusion from an inhomogeneous medium that contains a defect-rich implanted region near-surface to a relatively pristine region below the implant range better explains the observed diffusion behavior than a conventional model that assumes a homogeneous medium. By analyzing the diffusion behavior using the Boltzmann–Matano method, we present a discussion of reduction in [V Ga ] by thermal annealing at 1000 °C, leading to a suppressed diffusion coefficient during subsequent high-temperature annealing relative to diffusion after 1420/1500 °C annealing alone. This effect holds potential for improvement in the precision of selectively doped regions for future applications based on the (Al)GaN material system. An improved field profile control in real devices can increase the breakdown and current-handling capabilities in power electronic applications.

Meyers, V.↗

DOE OSTI2023

Surface stability and H adsorption and diffusion near surfaces of W borides: a first-principles study

Abstract Understanding the behavior of tungsten boride (W x B y ) surfaces in a fusion reactor environment is an important topic since boronization is a common wall conditioning method used in fusion Tokamaks. We report the results of density functional theory calculations that investigate the surface stability of W x B y (tetragonal I4 1 /amd -WB, hexagonal P 6 3 / mmc -WB 2 and tetragonal I4/m- W 2 B) with low-index orientations, as well as hydrogen (H) energetics near W x B y surfaces. For single element terminated W x B y surfaces, B terminated surfaces are more energetically stable than W terminated as a result of significant reconstruction of B. The H surface adsorption energy and activation energy of H diffusion penetration below W x B y surfaces are mainly related to the outer termination. Specifically, the WB(001) surface terminated with two B layers, referred to as WB(001)-T BB , has higher H adsorption affinity and lower H diffusivity on this surface than other terminations, which is controlled by the significant charge transfer from B to H. However, B atoms on the WB 2 (0001)-T BB surface decrease both H adsorption and diffusivity on the surface, but enhance H diffusion below the surface in comparison to W terminated WB 2 (0001) surface. H would be trapped and diffuse within atomic surface gaps on the WB 2 ( 2 1 ˉ 1 ˉ 0 ) surface, while H below the surface layer would jump along the [0001] direction rather than diffuse into bulk. The surface diffusion activation energy of H on the W 2 B(001) surface slightly varies with terminations. Once H crosses the surface layer of W 2 B(001) with either termination, it prefers to diffuse into the bulk, or back towards the surface, rather than move parallel to the surface. Interestingly, WB 2 (0001) and WB 2 ( 2 1 ˉ 1 ˉ 0 ) surfaces will have relatively higher H retention than the other W x B y surfaces evaluated in this work.

Yang, L. (ORCID:0000000322166071)↗

DOE OSTI2021

Is diffuse intracluster light a good tracer of the galaxy cluster matter distribution?

We explore the relation between diffuse intracluster light (central galaxy included) and the galaxy cluster (baryonic and dark) matter distribution using a sample of 528 clusters at 0.2 ≤ z ≤ 0.35 found in the Dark Energy Survey (DES) Year 1 data. The surface brightness of the diffuse light shows an increasing dependence on cluster total mass at larger radius, and appears to be self-similar with a universal radial dependence after scaling by cluster radius. We also compare the diffuse light radial profiles to the cluster (baryonic and dark) matter distribution measured through weak lensing and find them to be comparable. The IllustrisTNG galaxy formation simulation, TNG300, offers further insight into the connection between diffuse stellar mass and cluster matter distributions – the simulation radial profile of the diffuse stellar component does not have a similar slope with the total cluster matter content, although that of the cluster satellite galaxies does. Regardless of the radial trends, the amount of diffuse stellar mass has a low-scatter scaling relation with cluster’s total mass in the simulation, out-performing the total stellar mass of cluster satellite galaxies. We conclude that there is no consistent evidence yet on whether or not diffuse light is a faithful radial tracer of the cluster matter distribution. Nevertheless, both observational and simulation results reveal that diffuse light is an excellent indicator of the cluster’s total mass.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Diffusion-limited relic particle production

We examine the thermal evolution of particle number densities in the early universe when the particles have a finite diffusion length. Assuming that annihilations are impossible when the mean separation of the particles is larger than their diffusion length, we derive a version of the Boltzmann equation for freeze out in this scenario and an approximate solution, accurate to better than 2%. The effect of a finite diffusion length is to increase the final relic freeze-out abundance over its corresponding value when diffusion effects are ignored. When diffusion is limited only by scattering off of the thermal background, and the annihilation cross section is bounded by unitarity, a significant effect on the freeze-out abundance requires a scattering cross section much larger than the annihilation cross section. A similar effect is demonstrated when the relic particles are produced via the freeze-in mechanism, but in this case the finite diffusion length is due to the scattering of particles that annihilate into the relic particle of interest. For freeze in, the effect of a finite diffusion length is to reduce the final relic particle abundance. The effects of a finite diffusion length are most important when the scattering cross section or the relic mass are very large. While we have not found a particularly compelling example where this would affect previous results, with the current interest in new dark matter candidates it could become an important consideration.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Phase-field model for diffusion-induced grain boundary migration: An application to battery electrodes

Diffusion-induced grain boundary migration is a phenomenon in which a grain boundary moves in response to the driving forces generated by diffusing solute species. For example, diffusing solute species change the atomic volume in the host material, either by filling a vacancy with a misfitting solute atom or by expanding host lattices through interstitial diffusion. These volume changes are inhomogeneous and are stored as elastic energy in the material that drives grain boundaries. In this paper, we introduce our previously developed Cahn-Hilliard–phase-field-crystal model (CH-PFC) as a computational tool to investigate diffusion-induced grain boundary migration in crystalline materials. This multiscale phase-field model couples the composition field of a diffusing species with the crystallographic texture of a host material. We apply the CH-PFC model to battery electrodes and investigate whether interstitial solute diffusion induces grain growth in FePO 4 /LiFePO 4 electrodes. To this end, we compute grain growth in 60 FePO 4 electrodes by conducting two parallel trials: In the first trial, we cycle the electrode and calculate diffusion-induced grain growth. In the second trial, we do not cycle the electrode and calculate curvature-driven grain growth. We find a statistically significant grain growth in the cycled electrodes and negligible grain growth in the noncycled electrodes. Altogether, we show that the CH-PFC model not only predicts electrode microstructures as a function of the Li composition, but also predicts the crystallographic features of an electrode during battery operation.

36 MATERIALS SCIENCE↗

DOE OSTI2014

Comparison of flat cleaved and cylindrical diffusing fibers as treatment sources for interstitial photodynamic therapy

Purpose: For interstitial photodynamic therapy (iPDT) of bulky tumors, careful treatment planning is required in order to ensure that a therapeutic dose is delivered to the tumor, while minimizing damage to surrounding normal tissue. In clinical contexts, iPDT has typically been performed with either flat cleaved or cylindrical diffusing optical fibers as light sources. Here, the authors directly compare these two source geometries in terms of the number of fibers and duration of treatment required to deliver a prescribed light dose to a tumor volume. Methods: Treatment planning software for iPDT was developed based on graphics processing unit enhanced Monte Carlo simulations. This software was used to optimize the number of fibers, total energy delivered by each fiber, and the position of individual fibers in order to deliver a target light dose (D{sub 90}) to 90% of the tumor volume. Treatment plans were developed using both flat cleaved and cylindrical diffusing fibers, based on tissue volumes derived from CT data from a head and neck cancer patient. Plans were created for four cases: fixed energy per fiber, fixed number of fibers, and in cases where both or neither of these factors were fixed. Results: When the number of source fibers was fixed at eight, treatment plans based on flat cleaved fibers required each to deliver 7180–8080 J in order to deposit 90 J/cm{sup 2} in 90% of the tumor volume. For diffusers, each fiber was required to deliver 2270–2350 J (333–1178 J/cm) in order to achieve this same result. For the case of fibers delivering a fixed 900 J, 13 diffusers or 19 flat cleaved fibers at a spacing of 1 cm were required to deliver the desired dose. With energy per fiber fixed at 2400 J and the number of fibers fixed at eight, diffuser fibers delivered the desired dose to 93% of the tumor volume, while flat cleaved fibers delivered this dose to 79%. With both energy and number of fibers allowed to vary, six diffusers delivering 3485–3600 J were required, compared to ten flat cleaved fibers delivering 2780–3600 J. Conclusions: For the same number of fibers, cylindrical diffusers allow for a shorter treatment duration compared to flat cleaved fibers. For the same energy delivered per fiber, diffusers allow for the insertion of fewer fibers in order to deliver the same light dose to a target volume.

62 RADIOLOGY AND NUCLEAR MEDICINE↗

DOE OSTI2017

Progress on Fabrication of Planar Diffusion Couples with Representative TRISO PyC/SiC Microstructure

Release of fission products from tristructural-isotropic (TRISO) coated particle fuel limits the fuel’s operational lifetime and creates potential safety and maintenance concerns. A need for diffusion analysis in representative TRISO layers exists to provide fuel performance models with high fidelity data to improve fuel performance and efficiency. An effort has been initiated to better understand fission product transport in, and release from, quality TRISO fuel by investigating diffusion couples with representative pyrocarbon (PyC) and silicon carbide (SiC). Here planar PyC/SiC diffusion couples are being developed with representative PyC/SiC layers using a fluidized bed chemical vapor deposition (FBCVD) system identical to those used to produce laboratory-scale TRISO fuel for the Advanced Gas Reactor Fuel Qualification and Development Program’s (AGR) first fuel irradiation. The diffusivity of silver, the silver and palladium system, europium, and strontium in the PyC/SiC will be studied at elevated temperatures and under high temperature neutron irradiation. The study also includes a comparative study of PyC/SiC diffusion couples with varying TRISO layer properties to understand the influence of SiC microstructure (grain size) and the PyC/SiC interface on fission product transport. The first step in accomplishing these goals is the development of the planar diffusion couples. The diffusion couple construction consists of multiple steps which includes fabrication of the primary PyC/SiC structures with targeted layer properties, introduction of fission product species and seal coating to create an isolated system. Coating development has shown planar PyC/SiC diffusion couples with similar properties to AGR TRISO fuel can be produced. A summary of the coating development process, characterization methods, and status are presented.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2019

Predictions of diffusion rates of large organic molecules in secondary organic aerosols using the Stokes–Einstein and fractional Stokes–Einstein relations

Information on the rate of diffusion of organic molecules within secondary organic aerosol (SOA) is needed to accurately predict the effects of SOA on climate and air quality. Diffusion can be important for predicting the growth, evaporation, and reaction rates of SOA under certain atmospheric conditions. Often, researchers have predicted diffusion rates of organic molecules within SOA using measurements of viscosity and the Stokes–Einstein relation ($D∝1/η$, where D is the diffusion coefficient and η is viscosity). However, the accuracy of this relation for predicting diffusion in SOA remains uncertain. Using rectangular area fluorescence recovery after photobleaching (rFRAP), we determined diffusion coefficients of fluorescent organic molecules over 8 orders in magnitude in proxies of SOA including citric acid, sorbitol,and a sucrose–citric acid mixture. These results were combined with literature data to evaluate the Stokes–Einstein relation for predicting the diffusion of organic molecules in SOA. Although almost all the data agree with the Stokes–Einstein relation within a factor of 10, a fractional Stokes–Einstein relation ($D∝1/η^ξ$) with ξ =0.93 is a better model for predicting the diffusion of organic molecules in the SOA proxies studied. In addition, based on the output from a chemical transport model, the Stokes–Einstein relation can overpredict mixing times of organic molecules within SOA by as much as 1 order of magnitude at an altitude of ~3 km compared to the fractional Stokes–Einstein relation with ξ =0.93. These results also have implications for other areas such as in food sciences and the preservation of biomolecules.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1972

Analytical investigation of conical diffusers

A study was made to determine if existing data for straight-channel single-plane divergence diffusers together with boundary-layer methods could be practically employed to predict conical diffuser performance in the high-speed flow regime. A semiempirical correlation technique, which uses the experimental performance of a straight-channel single-plane divergence diffuser, and a momentum integral boundary-layer method were employed to predict conical diffuser performance. Neither of these methods is applicable when the boundary layer separates. The momentum integral method was used to study the relative importance of various boundary layer parameters in correlating unstalled diffuser performance. Other investigators have found that the best diffuser performance occurs when the diffuser is operating in the transitory stall regime. Since the analytical methods presented in this report are not applicable after the point of boundary-layer separation, a criterion was established from examination of experimental tests which allows the designer to ascertain approximate diffuser performance in the transitory stall regime.

Means, J. L.↗

NASA NTRS1973

Performance of a low-pressure-ratio centrifugal compressor with four diffuser designs

A low-pressure-ratio centrifugal compressor was tested with four different diffuser configurations. One diffuser had airfoil vanes. Two were pipe diffusers. One pipe diffuser had 7.5 deg cone diffusing passages. The other had trumpet-shaped passages designed for linear static-pressure rise from throat to exit. The fourth configuration had flat vanes with elliptical leading edges similar to those of pipe diffusers. The side walls were contoured to produce a linear pressure rise. Peak compressor efficiencies were 0.82 with the airfoil vane and conical pipe diffusers, 0.80 with the trumpet, and 0.74 with the flat-vane design. Surge margin and useful range were greater for the airfoil-vane diffuser than for the other three.

Klassen, H. A.↗

NASA NTRS1980

The contribution of the diffuse light component to the topographic effect on remotely sensed data

The topographic effect is measured by the difference between the global radiance from inclined surfaces as a function of their orientation relative to the sensor position and light source. The short wave radiant energy incident on a surface is composed of direct sunlight, scattered skylight, and light reflected from surrounding terrain. The latter two components are commonly known as the diffuse component. The contribution of the diffuse light component to the topographic effect was examined and the significance of this diffuse component with respect to two direct radiance models was assessed. Diffuse and global spectral radiances were measured for a series of slopes and aspects of a uniform and surface in the red and photographic infrared parts of the spectrum, using a nadir pointing two channel handheld radiometer. The diffuse light was found to produce a topographic effect which varied from the topographic effect for direct light. The topographic effect caused by diffuse light was found to increase slightly with solar elevation and wavelength for the channels examined. The correlations between data derived from two simple direct radiance simulation models and the field data were not significantly affected when the diffuse component was removed from the radiances. Radiances from a 60 percent reflective surface, assuming no atmospheric path radiance, the diffuse light topographic effect contributed a maximum range of 3 pixel values in simulated LANDSAT data from all aspects with slopes up to 30 degrees.

Justice, C.↗

NASA NTRS1987

Computation of diffuse sky irradiance from multidirectional radiance measurements

Accurate determination of the diffuse solar spectral irradiance directly above the land surface is important in characterizing the reflectance properties of these surfaces, especially vegetation canopies. This determination is also needed to infer the net radiation budget of the earth-atmosphere system above these surfaces. An algorithm is developed here for the computation of hemispheric diffuse irradiance using the measurements from an instrument called PARABOLA, which rapidly measures upwelling and downwelling radiances in three selected wavelength bands. The validity of the algorithm is established from simulations. The standard reference data set of diffuse radiances of Dave (1978), obtained by solving the radiative transfer equation numerically for realistic atmospheric models, is used to simulate PARABOLA radiances. Hemispheric diffuse irradiance is estimated from a subset of simulated radiances by using the algorithm described. The algorithm is validated by comparing the estimated diffuse irradiance with the true diffuse irradiance of the standard data set. The validations include sensitivity studies for two wavelength bands (visible, 0.65-0.67 micron; near infrared, 0.81-0.84 micron), different atmospheric conditions, solar elevations, and surface reflectances. In most cases the hemispheric diffuse irradiance computed from simulated PARABOLA radiances and the true irradiance obtained from radiative transfer calculations agree within 1-2 percent. This technique can be applied to other sampling instruments designed to estimate hemispheric diffuse sky irradiance.

Ahmad, Suraiya P.↗

NASA NTRS1991

Diffusion mechanisms in Ir-coated Re for high-temperature, radiation-cooled rocket thrusters

Materials used for radiation-cooled rocket thrusters must be capable of surviving under extreme conditions of high temperatures and oxidizing environments. Thruster chambers were developed using chemical-vapor-deposited (CVD) Re coated with CVD Ir on the inside surface which is exposed to hot combustion gases. Ir serves as an oxidation barrier protecting the Re which maintains structural integrity at high temperatures. In order to predict and extend the performance limits of these Ir-coated Re thrusters, the diffusion kinetics of CVD materials at temperature are studied. Thruster end ring sections were examined using electron microprobe analysis both before and after exposure to high temperature vacuum environments. The resulting elemental maps for Re, Ir, and Mo in the near-surface region allow identification of diffusion mechanisms operating at these temperatures. Line scans for Ir and Re were fit using a diffusion model to extract relevant diffusion constants. The fastest diffusion process is seen to be grain boundary diffusion with Re diffusing down grain boundaries in the Ir overlayer. The measured dependence of the diffusion rate on temperature will allow prediction of operating lifetimes for these thrusters.

Hamilton, J. C.↗

NASA NTRS1956

An analog study of a shock-position diffuser control on a supersonic turbojet engine

Shock position was assumed to be controlled by air bleed through a diffuser-exit bypass door operated by a second-order servomotor. Pressure disturbance resulting from changes in bypass-door position had whereas a disturbance in engine speed had a large effect on diffuser pressure and could impose severe requirements on diffuser control response. The controlled diffuser response improved with faster servomotor response and smaller diffuser dead time and lag. The effect of diffuser dead time on diffuser response was greater than the effect of diffuser lag.

Novik, David↗

NASA NTRS1995

Radiant extinction of gaseous diffusion flames

The absence of buoyancy-induced flows in microgravity significantly alters the fundamentals of many combustion processes. Substantial differences between normal-gravity and microgravity flames have been reported during droplet combustion, flame spread over solids, candle flames, and others. These differences are more basic than just in the visible flame shape. Longer residence time and higher concentration of combustion products create a thermochemical environment which changes the flame chemistry. Processes such as flame radiation, that are often ignored under normal gravity, become very important and sometimes even controlling. This is particularly true for conditions at extinction of a microgravity diffusion flame. Under normal-gravity, the buoyant flow, which may be characterized by the strain rate, assists the diffusion process to transport the fuel and oxidizer to the combustion zone and remove the hot combustion products from it. These are essential functions for the survival of the flame which needs fuel and oxidizer. Thus, as the strain rate is increased, the diffusion flame which is 'weak' (reduced burning rate per unit flame area) at low strain rates is initially 'strengthened' and eventually it may be 'blown-out'. Most of the previous research on diffusion flame extinction has been conducted at the high strain rate 'blow-off' limit. The literature substantially lacks information on low strain rate, radiation-induced, extinction of diffusion flames. At the low strain rates encountered in microgravity, flame radiation is enhanced due to: (1) build-up of combustion products in the flame zone which increases the gas radiation, and (2) low strain rates provide sufficient residence time for substantial amounts of soot to form which further increases the flame radiation. It is expected that this radiative heat loss will extinguish the already 'weak' diffusion flame under certain conditions. Identifying these conditions (ambient atmosphere, fuel flow rate, fuel type, etc.) is important for spacecraft fire safety. Thus, the objective is to experimentally and theoretically investigate the radiation-induced extinction of diffusion flames in microgravity and determine the effect of flame radiation on the 'weak' microgravity diffusion flame.

Atreya, Arvind↗

NASA NTRS1998

Comments on the Diffusive Behavior of Two Upwind Schemes

The diffusive characteristics of two upwind schemes, multi-dimensional fluctuation splitting and locally one-dimensional finite volume, are compared for scalar advection-diffusion problems. Algorithms for the two schemes are developed for node-based data representation on median-dual meshes associated with unstructured triangulations in two spatial dimensions. Four model equations are considered: linear advection, non-linear advection, diffusion, and advection-diffusion. Modular coding is employed to isolate the effects of the two approaches for upwind flux evaluation, allowing for head-to-head accuracy and efficiency comparisons. Both the stability of compressive limiters and the amount of artificial diffusion generated by the schemes is found to be grid-orientation dependent, with the fluctuation splitting scheme producing less artificial diffusion than the finite volume scheme. Convergence rates are compared for the combined advection-diffusion problem, with a speedup of 2.5 seen for fluctuation splitting versus finite volume when solved on the same mesh. However, accurate solutions to problems with small diffusion coefficients can be achieved on coarser meshes using fluctuation splitting rather than finite volume, so that when comparing convergence rates to reach a given accuracy, fluctuation splitting shows a speedup of 29 over finite volume.

Wood, William A.↗