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NASA NTRS2016

Probabilistic Modeling of Aircraft Trajectories for Dynamic Separation Volumes

With a proliferation of new and unconventional vehicles and operations expected in the future, the ab initio airspace design will require new approaches to trajectory prediction for separation assurance and other air traffic management functions. This paper presents an approach to probabilistic modeling of the trajectory of an aircraft when its intent is unknown. The approach uses a set of feature functions to constrain a maximum entropy probability distribution based on a set of observed aircraft trajectories. This model can be used to sample new aircraft trajectories to form an ensemble reflecting the variability in an aircraft's intent. The model learning process ensures that the variability in this ensemble reflects the behavior observed in the original data set. Computational examples are presented.

Lewis, Timothy A.↗

NASA NTRS2018

Hybrid Reduced Order Model for N2-N2 Interactions for Application to Dissociation and Energy Transfer Processes

Recent work in the aerothermodynamics community has focused on the development of reduced-order models for thermo-chemical non-equilibrium which avoid the restrictive assumptions of multi-temperature models and the prohibitive cost associated with State-to-State (StS) models. In the present work, this is accomplished by lumping energy states together and assuming groups of states are roughly in equilibrium. As a result, the non-equilibrium behavior of a gas can be captured at a reduced computational cost from a full StS simulation. In this work, we present a hybrid grouping model for studying energy transfer and dissociation in a mixture of nitrogen molecules due to N2-N2 reactions. This is accomplished by making use of a grouping strategy informed by data from the N2-N StS kinetic data. However, due to the massive computational cost associated with constructing StS data for the N2-N2 system, the kinetic data for the hybrid grouping model are calculated using the quasi-classical trajectory (QCT) method by sampling states for trajectory within the groups. This general framework is called the Maximum-Entropy Quasi-Classical Trajectory (ME-QCT) method. The primary challenge associated with this method is that rates for reverse grouped reactions cannot be obtained through detailed balance at a group level, due to the variation of group internal temperatures. To construct the full model for N2-N2 grouped kinetics using the ME-QCT method, detailed balance is invoked at the microscopic level, allowing for the calculation of the full kinetic data from QCT. Results will be presented using the full ME-QCT model for the N2-N2 system in an isothermal and isochoric reactor simulation. In addition, simple CFD test cases for a one-dimensional standing shock and for a quasi-one-dimensional nozzle will be used for demonstration of the ME-QCT method. This method allows for the calculation of non-equilibrium behavior for the N2-N2 system without the prohibitive cost of a full StS simulation. Moreover, it enables the construction of a unified model for the dissociating and recombining non-equilibrium flows.

nonequilibrium modeling↗

NASA NTRS2022

Bryce Canyon Water Resources: Monitoring Vegetation Health and Water Availability in Bryce Canyon National Park for Drought Stress Mitigation Planning

Bryce Canyon National Park is home to groundwater-dependent ecosystems (GDEs) that are threatened by a multidecadal drought and increased groundwater extraction due to a spike in tourism. These ecosystems contain unique species that are only found in areas where near-surface groundwater is present, such as aspen groves and fens. These species contribute to the high biodiversity found in Bryce Canyon, which boosts an ecosystem’s productivity and the services it provides to the park. Unfortunately, many of these GDEs are too small to identify with traditional Earth observation platforms and are difficult to physically reach for monitoring purposes. This project partnered with the National Park Service to identify springs and seeps as a proxy for GDEs within Bryce Canyon from 2013–2022. Furthermore, this project tested the feasibility of various methods to detect and monitor springs and seeps and therefore facilitate the partner’s efforts to conserve these ecologically valuable GDEs in Bryce Canyon. The team mapped groundwater discharge with high resolution National Agriculture Imagery Program (NAIP) and assessed park vegetation trends with Landsat 8 Operational Land Imager (OLI) and PlanetScope imagery. In-situ precipitation data and the Western Land Data Assimilation System (WLDAS) were used to produce time series of climatic variables. Seeps and spring locations were predicted using random forest classification and maximum entropy machine learning models.

Groundwater dependent ecosystems↗

NASA NTRS2019

Size and Structure of the Sequence Space of Repeat Proteins

The coding space of protein sequences is shaped by evolutionary constraints set by requirements of function and stability. We show that the coding space of a given protein family— the total number of sequences in that family—can be estimated using models of maximum entropy trained on multiple sequence alignments of naturally occurring amino acid sequences. We analyzed and calculated the size of three abundant repeat proteins families, whose members are large proteins made of many repetitions of conserved portions of *30 amino acids. While amino acid conservation at each position of the alignment explains most of the reduction of diversity relative to completely random sequences, we found that correlations between amino acid usage at different positions significantly impact that diversity. We quantified the impact of different types of correlations, functional and evolutionary, on sequence diversity. Analysis of the detailed structure of the coding space of the families revealed a rugged landscape, with many local energy minima of varying sizes with a hierarchical structure, reminiscent of frustrated energy landscapes of spin glass in physics. This clustered structure indicates a multiplicity of subtypes within each family and suggests new strategies for protein design.

Jacopo Marchi↗

DOE OSTI2011

Charmonium spectral functions with the variational method in zero and finite temperature lattice QCD

We propose a method to evaluate spectral functions on the lattice based on a variational method. On a lattice with a finite spatial extent, spectral functions consist of discrete spectra only. Adopting a variational method, we calculate the locations and the heights of spectral functions at low-lying discrete spectra. We first test the method in the case of analytically solvable free Wilson quarks at zero and finite temperatures and confirm that the method well reproduces the analytic results for low-lying spectra. We find that we can systematically improve the results by increasing the number of trial states. We then apply the method to calculate the charmonium spectral functions for S and P-wave states at zero-temperature in quenched QCD and compare the results with those obtained using the conventional maximum entropy method (MEM). The results for the ground state are consistent with the location and the area of the first peak in spectral functions from the MEM, while the variational method leads to a mass which is closer to the experimental value for the first excited state. We also investigate the temperature dependence of the spectral functions for S-wave states below and above T{sub c}. We obtain no clear evidences for dissociation of J/{psi} and {eta}{sub c} up to 1.4 T{sub c}.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2012

Atmospheric PSF Interpolation for Weak Lensing in Short Exposure Imaging Data

A main science goal for the Large Synoptic Survey Telescope (LSST) is to measure the cosmic shear signal from weak lensing to extreme accuracy. One difficulty, however, is that with the short exposure time ({approx_equal}15 seconds) proposed, the spatial variation of the Point Spread Function (PSF) shapes may be dominated by the atmosphere, in addition to optics errors. While optics errors mainly cause the PSF to vary on angular scales similar or larger than a single CCD sensor, the atmosphere generates stochastic structures on a wide range of angular scales. It thus becomes a challenge to infer the multi-scale, complex atmospheric PSF patterns by interpolating the sparsely sampled stars in the field. In this paper we present a new method, psfent, for interpolating the PSF shape parameters, based on reconstructing underlying shape parameter maps with a multi-scale maximum entropy algorithm. We demonstrate, using images from the LSST Photon Simulator, the performance of our approach relative to a 5th-order polynomial fit (representing the current standard) and a simple boxcar smoothing technique. Quantitatively, psfent predicts more accurate PSF models in all scenarios and the residual PSF errors are spatially less correlated. This improvement in PSF interpolation leads to a factor of 3.5 lower systematic errors in the shear power spectrum on scales smaller than {approx} 13, compared to polynomial fitting. We estimate that with psfent and for stellar densities greater than {approx_equal}1/arcmin{sup 2}, the spurious shear correlation from PSF interpolation, after combining a complete 10-year dataset from LSST, is lower than the corresponding statistical uncertainties on the cosmic shear power spectrum, even under a conservative scenario.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

MULTI-SCALE MODELING AND APPROXIMATION ASSISTED OPTIMIZATION OF BARE TUBE HEAT EXCHANGERS

Air-to-refrigerant heat exchangers are very common in air-conditioning, heat pump and refrigeration applications. In these heat exchangers, there is a great benefit in terms of size, weight, refrigerant charge and heat transfer coefficient, by moving from conventional channel sizes (~ 9mm) to smaller channel sizes (< 5mm). This work investigates new designs for air-to-refrigerant heat exchangers with tube outer diameter ranging from 0.5 to 2.0mm. The goal of this research is to develop and optimize the design of these heat exchangers and compare their performance with existing state of the art designs. The air-side performance of various tube bundle configurations are analyzed using a Parallel Parameterized CFD (PPCFD) technique. PPCFD allows for fast-parametric CFD analyses of various geometries with topology change. Approximation techniques drastically reduce the number of CFD evaluations required during optimization. Maximum Entropy Design method is used for sampling and Kriging method is used for metamodeling. Metamodels are developed for the air-side heat transfer coefficients and pressure drop as a function of tube-bundle dimensions and air velocity. The metamodels are then integrated with an air-to-refrigerant heat exchanger design code. This integration allows a multi-scale analysis of air-side performance heat exchangers including air-to-refrigerant heat transfer and phase change. Overall optimization is carried out using a multi-objective genetic algorithm. The optimal designs found can exhibit 50 percent size reduction, 75 percent decrease in air side pressure drop and doubled air heat transfer coefficients compared to a high performance compact micro channel heat exchanger with same capacity and flow rates.

CFD↗

DOE OSTI2014

SIMPLIFIED PREDICTIVE MODELS FOR CO₂ SEQUESTRATION PERFORMANCE ASSESSMENT RESEARCH TOPICAL REPORT ON TASK #3 STATISTICAL LEARNING BASED MODELS

We compare two approaches for building a statistical proxy model (metamodel) for CO₂ geologic sequestration from the results of full-physics compositional simulations. The first approach involves a classical Box-Behnken or Augmented Pairs experimental design with a quadratic polynomial response surface. The second approach used a space-filling maxmin Latin Hypercube sampling or maximum entropy design with the choice of five different meta-modeling techniques: quadratic polynomial, kriging with constant and quadratic trend terms, multivariate adaptive regression spline (MARS) and additivity and variance stabilization (AVAS). Simulations results for CO₂ injection into a reservoir-caprock system with 9 design variables (and 97 samples) were used to generate the data for developing the proxy models. The fitted models were validated with using an independent data set and a cross-validation approach for three different performance metrics: total storage efficiency, CO₂ plume radius and average reservoir pressure. The Box-Behnken–quadratic polynomial metamodel performed the best, followed closely by the maximin LHS–kriging metamodel.

Mishra, Srikanta↗

DOE OSTI2015

System for uncollimated digital radiography

The inversion algorithm based on the maximum entropy method (MEM) removes unwanted effects in high energy imaging resulting from an uncollimated source interacting with a finitely thick scintillator. The algorithm takes as input the image from the thick scintillator (TS) and the radiography setup geometry. The algorithm then outputs a restored image which appears as if taken with an infinitesimally thin scintillator (ITS). Inversion is accomplished by numerically generating a probabilistic model relating the ITS image to the TS image and then inverting this model on the TS image through MEM. This reconstruction technique can reduce the exposure time or the required source intensity without undesirable object blurring on the image by allowing the use of both thicker scintillators with higher efficiencies and closer source-to-detector distances to maximize incident radiation flux. The technique is applicable in radiographic applications including fast neutron, high-energy gamma and x-ray radiography using thick scintillators.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2015

Design and Validation Testing of TruckScan to Assay Large Sacks of Fukushima Radioactive Debris on a Truck

As a result of the March 2011 earthquake and resulting tsunami in Japan, there was a serious accident at the Fukushima Dai-ichi Nuclear Power Plant. This accident has contaminated soil and vegetation in a wide area around the plant. Decontamination projects over the last 4 years have resulted in large numbers of 1 cubic meter canvas bags of debris, commonly called Super Sacks [SS]. These are currently stored nearby where they were generated, but starting in 2015, they will be moved to various Interim Storage Facilities [ISF]. Trucks will typically carry 8-20 of these SSs. When the trucks arrive at the ISF they need to be rapidly sorted into groups according to radioactivity level, for efficient subsequent processing. Canberra Industries, Inc. [CI] has designed a new truck monitoring system 'TruckScan' for use at these ISFs. The TruckScan system must measure the entire truck loaded with multiple closely packed SSs, and generate a nuclide specific assay report showing the radioactivity in each individual SS. The Canberra-Japan office, along with Obayashi Corporation have performed validation testing to demonstrate to the regulatory authorities that the proposed technique was sufficiently accurate. These validation tests were conducted at a temporary storage area in Fukushima prefecture. Decontaminated waste of various representative types and of various levels of radioactivity was gathered and mixed to create homogeneous volumes. These volumes were sampled multiple times and assayed with laboratory HPGe detectors to determine the reference concentration of each pile. Multiple SSs were loaded from each pile. Some of the SSs were filled 50% full, others 75% full, and others 100% full, to represent the typical loading configuration of the existing SSs in the field. The content of the SSs are either sand, soil, or vegetation with densities ranging from 0.3 g/cc - 1.6 g/cc. These SSs with known concentrations of Cs-134 and Cs-137 were then loaded onto trucks in a variety of configurations, typical of how they might be on the real trucks. A partial system was installed at the site and used to assay these trucks with the various loading configurations. Whereas the full system will have 8 collimated 3 x 3'' NaI detectors, the test system only had two detectors; therefore the truck was moved and counted 4 times. The data were acquired and analyzed with the Canberra Genie software to determine the peak counts for both Cesium nuclides. That data was then analyzed with a prototype version of a Maximum Entropy algorithm, to determine the individual SS activity. The goal of the validation tests was to demonstrate that the system could detect 1000 Bq/kg in 15 seconds, and to determine how accurately it could quantify individual SSs. The validation tests demonstrated that the product would perform as predicted. The TruckScan results were consistent with the sample assay results [y = 1.0029 x, R{sup 2} = 0.914]. The Total Propagated Uncertainty, including both uncertainties from these tests and others that were estimated but not tested was 16.6% percent. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2015

Validation testing of Canberra-Obayashi TruckScan calculation method - 15409

Canberra Industries, Inc. [CI] has designed a new truck monitoring system 'TruckScan' for Interim Storage Facilities [ISF] in Japan. CI and Canberra Japan K.K. [CJKK] have performed validation testing of it with Obayashi Corporation at a temporary storage area in Fukushima prefecture. Decontaminated waste was put into flexible containers called Super Sacks [SS]. The customer desires to quantify the Cs-137 and Cs-134 content of each individual SS, while a group of them are on a truck. This would be done when leaving a temporary waste consolidation area and entering an ISF. The content of SSs is some combination of sand, soil, and vegetation with densities ranging from 0.3 g/cc - 1.6 g/cc. The typical weight of the trucks will be approximately 10 tons, but can vary between 4 and 20 tons. The system must be sensitive enough to detect 100 Bq/kg in 10 - 30 seconds but still have enough dynamic range to measure 1,000,000 Bq/kg material. The system will be operated in an outdoor environment. The desire to separately quantify Cs-137 and Cs-134 favors the use of a spectroscopic system as a solution. The full-scale TruckScan will consist of eight 3x3' NaI detectors, each in a lead shield with a collimated view of the truck. Four detectors are on each side of the truck, at about 1 meter from the truck, spaced at equal distances. These NaI detectors and collimators were calibrated by In Situ Object Counting System (ISOCS) mathematical efficiency calculation tool. The truck stops in-between the two sets of detectors for the short measurement period - typically 15 seconds. The special software performs gamma spectroscopy on each of the 8 spectra, and then decodes the results to determine the activity in each of the 8 [typically] SSs using a Maximum Entropy Analysis Method. Validation testing was done by using SSs filled with known material types and known concentrations of material. The results of the TruckScan assay indicate good accuracy for the wide range of conditions. In spite of the conditions being more severe than normal operations, the combined standard deviation was 20.1% for the 6 detector version and 16.6% for the 8 detector version TruckScan. When the TruckScan results were compared to the known concentrations, there was minimal bias and good correlation [y = 1.0029x, R{sup 2} = 0.914]. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2020

The influence of Au substitution and hydrostatic pressure on the phase transitions and magnetocaloric properties of MnCoGe alloys

In this work, the phase transitions and magnetocaloric properties of Mn 1--x Au x CoGe (0≤x≤0.025) alloys were studied as a function of concentration x and applied hydrostatic pressure. The increasing substitution of Au for Mn results in the decrease of the first-order martensitic transition temperature, and this first-order martensitic transition was ultimately converted to a second-order magnetic transition when the Au substitution (x) reached 0.025. The magnitudes of the maximum magnetic entropy changes increased when the magnetic and structural transitions were coupled, which occurred for 0.005≤x≤0.020. The largest maximum magnetic entropy change for a field change of μ 0 ΔH=7 T was 33.1J/kgK for the sample with x=0.020. Similar to the effect of Au substitution, the first-order martensitic transition temperature initially decreased, and then converted to second order, when the applied hydrostatic pressure reached a large enough value. Interestingly, both Au substitution and pressure application cause a volume reduction and, in both cases, the first-order martensitic transition temperature initially reduced and then converted to second-order. These results suggest two different methods of tuning the transition temperatures in these magnetocaloric materials. One can either apply hydrostatic pressure and temporarily adjust the transition temperatures or modify the composition chemically and permanently change the transition temperatures.

36 MATERIALS SCIENCE↗

DOE OSTI2015

Sample dependence of giant magnetocaloric effect in a cluster-glass system Ho{sub 5}Pd{sub 2}

In order to investigate the effect of vacancy on the magnetocaloric effect in Ho{sub 5}Pd{sub 2}, we have carried out X-ray diffraction, magnetization, and specific heat measurements in the rare-earth intermetallic compound Ho{sub 5+x}Pd{sub 2}(−0.4 ≤ x ≤ 0.4). The maximum magnetic entropy change −ΔS{sub m}{sup max}, the maximum adiabatic temperature change ΔT{sub ad}{sup max}, and the relative cooling power of Ho{sub 5+x}Pd{sub 2} take large values at x = 0−0.4 for the field change of 5 T. The paramagnetic Curie temperature θ{sub p} increases with an increase of x. This fact suggests that the enhancement of ferromagnetic coupling among the correlated spins leads to the increase of magnetocaloric effect.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

Maximum Kolmogorov-Sinai Entropy Versus Minimum Mixing Time in Markov Chains

We establish a link between the maximization of Kolmogorov Sinai entropy (KSE) and the minimization of the mixing time for general Markov chains. Since the maximisation of KSE is analytical and easier to compute in general than mixing time, this link provides a new faster method to approximate the minimum mixing time dynamics. It could be interesting in computer sciences and statistical physics, for computations that use random walks on graphs that can be represented as Markov chains.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

High Magnetic Anisotropy and Magnetocaloric Effects in Single-Crystal Cr 2 Te 3

Here, we report a systematic investigation of anisotropic magnetocaloric effects in single-crystal Cr 2 Te 3 . Single-crystal samples are synthesized by chemical vapor transport and characterized by X-ray and Laue diffraction methods. The maximum magnetic entropy change –ΔS M max is 4.50 J kg –1 K –1 for the easy c-axis (3.36 J kg –1 K –1 for the hard axis along ab-plane), and the relative cooling power (RCP) is 296.7 J kg –1 for the easy c-axis (183.84 J kg –1 for the hard axis along ab-plane) for a magnetic field change of 9 T near the Curie temperature. The magneto-crystalline anisotropy constant K u is estimated to be 580.12 kJ m –3 at 140 K, decreasing to 148.60 kJ m –3 at 168 K. Meanwhile, the maximum of the rotational magnetic entropy change –ΔS M R (T, H) between the c-axis and the ab-plane is about 1.14 J kg –1 K –1 for magnetic-field change of 9 T. The critical exponents are estimated by analyzing magnetocaloric effects, which indicate a 2D-Ising type magnetic system. The accuracy of estimated critical exponents is verified by scaling analysis. The maximum magnetic entropy change –ΔS M max ≈ 5.25 J kg –1 K –1 (along the c-axis) and the corresponding adiabatic temperature change ΔT ad ≈ 3.31 K (along the c-axis) are estimated by analyzing heat capacity measurements with a magnetic field up to 9 T.

36 MATERIALS SCIENCE↗

DOE OSTI2015

Critical behavior and magnetocaloric effect of Pr{sub 1−x}Ca{sub x}MnO{sub 3}

The critical behavior of Pr{sub 1−x}Ca{sub x}MnO{sub 3} samples with x = 0.25, 0.27, and 0.29 has been investigated. Detailed analyses of magnetic-field dependences of magnetization at temperatures around the paramagnetic-ferromagnetic transition, M(H, T), reveal that the samples undergo a second-order magnetic phase transition. The Arrott plot method predicts the values of critical parameters to be T{sub C} ≈ 118 K, β = 0.351 ± 0.003, γ = 1.372 ± 0.002, and δ = 4.90 ± 0.02 for x = 0.25; T{sub C} ≈ 116 K, β = 0.362 ± 0.002, γ = 1.132 ± 0.004, and δ = 4.09 ± 0.03 for x = 0.27; and T{sub C} ≈ 110 K, β = 0.521 ± 0.002, γ = 0.912 ± 0.005, and δ = 2.71 ± 0.02 for x = 0.29. The values of β = 0.351 (for x = 0.25) and β = 0.362 (for x = 0.27) are close to the value β = 0.365 expected for the 3D Heisenberg model, proving an existence of short-range ferromagnetic interactions in these samples. A slight increase in Ca-doping content (x = 0.29) leads to the shift of the β value (=0.521) towards that of the mean-field theory (with β = 0.5) characteristic of long-range ferromagnetic interactions. The samples also exhibit a magnetocaloric effect: around T{sub C} of Pr{sub 1−x}Ca{sub x}MnO{sub 3} compounds, magnetic-entropy change reaches the maximum values of about 5.0, 4.1, and 2.5 J kg{sup −1} K{sup −1} for x = 0.25, 0.27, and 0.29, respectively, under an applied-field change of 50 kOe. Magnetic-field dependences of the maximum magnetic-entropy change (ΔS{sub max}) obey a power law |ΔS{sub max}(H)| ∝ H{sup n}, where exponent values n = 0.68–0.74 are close to those obtained from the theoretical relation n = 1 + (β − 1)/(β + γ)

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

Investigation of Magnetic, Magnetocaloric, and Critical Properties of La{sub 0.5}Ba{sub 0.5}MnO{sub 3} Manganite

We present an extensive study of the magnetic properties of a novel La{sub 0.5}Ba{sub 0.5}MnO{sub 3} perovskite material prepared by the hydrothermal method. The explored sample was structurally studied by the x-ray diffraction (XRD) method which confirms the formation of a pure cubic phase of a perovskite structure with Pm3m space group. The magnetic properties were probed by employing temperature M (T) and external magnetic field M (μ{sub o}H) dependence of magnetization measurements. A magnetic phase transition from ferromagnetic to paramagnetic phase occurs at 339 K in this sample. The maximum magnetic entropy change (|ΔS{sub M}{sup max}|) took a value of 1.4 J kg{sup − 1} K{sup − 1} at the applied magnetic field of 4.0 T for the explored sample and has also been found to occur at Curie temperature (T{sub C}). This large entropy change might be instigated from the abrupt reduction of magnetization at T{sub C}. The magnetocaloric effect (MCE) is maximum at T{sub C} as represented by M (μ{sub o}H) isotherms. The relative cooling power (RCP) is 243.2 J kg{sup − 1} at μ{sub o}H = 4.0 T. Moreover, the critical properties near T{sub C} have been probed from magnetic data. The critical exponents δ, β, and γ with values 3.82, 0.42, and 1.2 are close to the values predicted by the 3D Ising model. Additionally, the authenticity of the critical exponents has been confirmed by the scaling equation of state and all data fall on two separate branches, one for T < T{sub C} and the other for T > T{sub C}, signifying that the critical exponents obtained in this work are accurate.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

Physics-based stabilized finite element approximations of the Poisson–Nernst–Planck equations

We present and analyze two stabilized finite element methods for solving numerically the Poisson–Nernst–Planck equations. The stabilization we consider is carried out by using a shock detector and a discrete graph Laplacian operator for the ion equations, whereas the discrete equation for the electric potential need not be stabilized. Discrete solutions stemmed from the first algorithm preserve both maximum and minimum discrete principles. For the second algorithm, its discrete solutions are conceived so that they hold discrete principles and obey an entropy law provided that an acuteness condition is imposed for meshes. Remarkably the latter is found to be unconditionally stable. We validate our methodology through transient numerical experiments that show convergence toward steady-state solutions.

97 MATHEMATICS AND COMPUTING↗