Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “direct simulation Monte Carlo methods”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 325 records · Page 18
All sourcesNASA NTRSDOE OSTI
NASA NTRS2016

Giant Planets in Reflected Light: What Science Can We Expect?

Interpreting the reflection spectra of cool giant planets will be a challenge. Spectra of such worlds are expected to be primarily shaped by scattering from clouds and hazes and punctuated by absorption bands of methane, water, and ammonia. While the warmest giants may be cloudless, their atmospheres will almost certainly sport substantial photochemical hazes. Furthermore the masses of most direct imaging targets will be constrained by radial velocity observations, their radii, and thus atmospheric gravity, will be imperfectly known. The uncertainty in planet radius and gravity will compound with uncertain aerosol properties to make estimation of key absorber abundances difficult. To address such concerns our group is developing atmospheric retrieval tools to constrain quantities of interest, particular gas mixing ratios. We have applied our Markov Chain Monte Carlo methods to simulated data of the quality expected from the WFIRST CGI instrument and found that given sufficiently high SNR data we can confidentially identify and constrain the abundance of methane, cloud top pressures, gravity, and the star-planet-observer phase angle. In my presentation I will explain the expected characteristics of cool extrasolar giant planet reflection spectra, discuss these and other challenges in their interpretation, and summarize the science results we can expect from direct imaging observations.

methane↗
NASA NTRS2004

A Massively Parallel Hybrid Dusty-Gasdynamics and Kinetic Direct Simulation Monte Carlo Model for Planetary Applications

In order to understand the global structure, dynamics, and physical and chemical processes occurring in the upper atmospheres, exospheres, and ionospheres of the Earth, the other planets, comets and planetary satellites and their interactions with their outer particles and fields environs, it is often necessary to address the fundamentally non-equilibrium aspects of the physical environment. These are regions where complex chemistry, energetics, and electromagnetic field influences are important. Traditional approaches are based largely on hydrodynamic or magnetohydrodynamic (MHD) formulations and are very important and highly useful. However, these methods often have limitations in rarefied physical regimes where the molecular collision rates and ion gyrofrequencies are small and where interactions with ionospheres and upper neutral atmospheres are important. At the University of Michigan we have an established base of experience and expertise in numerical simulations based on particle codes which address these physical regimes. The Principal Investigator, Dr. Michael Combi, has over 20 years of experience in the development of particle-kinetic and hybrid kinetichydrodynamics models and their direct use in data analysis. He has also worked in ground-based and space-based remote observational work and on spacecraft instrument teams. His research has involved studies of cometary atmospheres and ionospheres and their interaction with the solar wind, the neutral gas clouds escaping from Jupiter s moon Io, the interaction of the atmospheres/ionospheres of Io and Europa with Jupiter s corotating magnetosphere, as well as Earth s ionosphere. This report describes our progress during the year. The contained in section 2 of this report will serve as the basis of a paper describing the method and its application to the cometary coma that will be continued under a research and analysis grant that supports various applications of theoretical comet models to understanding the inner comae of comets (grant NAGS- 13239 from the Planetary Atmospheres program).

Combi, Michael R.↗
NASA NTRS1993

Comparison of remote sensing measurements with a two-scale polarimetric emission and scattering model for sea surfaces

Recently, it has been observed that the brightness temperatures of sea surfaces correlate with the azimuth angle of the ocean wind vector, indicating that ocean wind direction can not only be retrieved from the microwave backscatter, but also from the brightness temperature measurements. In this paper, comparison of the theoretical result calculated from a two-scale emission and scattering model with the Seasat and SSM/I model functions and aircraft measurements is presented and potential applications of microwave polarimetry are discussed. In our two-scale model, the modified reflectivities of large scale surfaces are calculated by extending the small perturbation method to the second order for small scale perturbation with anisotropic directional spectrum. It was found that the modified reflectivities derived from the second-order scattered field agree excellently well with the results obtained from a Monte Carlo simulation technique which numerically calculates the polarimetric reflectivities of one-dimensional random rough surfaces with a power-law spectrum. Without the second-order correction, the modified reflectivities of the rough surfaces are significantly over-estimated and sign errors are observed in the third and fourth Stokes parameters for thermal emissions. The surface spectrum parameters and two-scale cutoff are selected so that the calculated scattering and emission signatures agree with the reported model functions for Seasat and SSM/I. Subsequently, the polarimetric signatures of sea surfaces are illustrated to indicate the possibility of reducing the number of azimuthal looks required for spaceborne sensors in the remote sensing of ocean wind by using the polarimetric information. Furthermore, it is found that contrary to the dependence of backscattering coefficients on incidence angles, polarimetric brightness temperatures display a stronger wind direction dependence in the near nadir-looking direction than away from nadir. Finally, we discuss the potential of a single-look polarimetric radiometer for wind retrieval.

Yueh, S. H.↗
DOE OSTI2020

Computational modeling of charge hopping dynamics along a disordered one-dimensional wire with energy gradients in quantum environments

This computational study investigates the effects of energy gradients on charge hopping dynamics along a one-dimensional chain of discrete sites coupled to quantum bath, which is modeled at the level of Pauli master equation (PME). This study also assesses the performance of different approximations for the hopping rates. Three different methods for solving the PME, a fourth order Runge–Kutta method, numerical diagonalization of the rate matrix followed by analytic propagation, and kinetic Monte Carlo simulation method, are tested and confirmed to produce virtually identical values of time dependent mean square displacement, diffusion constant, and mobility. Five different rate expressions, exact numerical evaluation of Fermi’s Golden Rule (FGR) rate, stationary phase interpolation (SPI) approximation, semiclassical approximation, classical Marcus rate, and Miller–Abrahams rate, are tested to help understand the effects of approximations in representing quantum environments in the presence of energy gradients. The results based on direct numerical evaluation of FGR rate exhibit transition from diffusive to non-diffusive behavior with the increase in the gradient and show that the charge transport in the quantum bath is more sensitive to the magnitude of the gradient and the disorder than in the classical bath. Among all the four approximations for the hopping rates, the SPI approximation is confirmed to work best overall. A comparison of two different methods to calculate the mobility identifies drift motion of the population distribution as the major source of non-diffusive behavior and provides more reliable information on the contribution of quantum bath.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Clear-sky detection for PV degradation analysis using multiple regression

A method is presented to detect clear-sky periods for plane-of-array irradiance time-averaged data that is based on the algorithm originally described by Reno and Hansen. Here we show this new method improves the state-of-the-art by providing accurate detection at longer data averaging intervals. Moreover, our new method detects clear periods in plane-of-array data, which is novel. The new method is developed by applying a Design of Experiment approach to optimize the parameters used in the Reno method, and Monte Carlo simulations are used to understand the robustness of the found parameters. Clear-sky detection accuracy is compared among four methods: the Reno method, the default clear-sky filter in RdTools, the Ellis method, and the method outlined in this work, using a hand-labeled two-year data set of 1-min plane-of-array irradiance for a fixed tilt system. The RdTools clear-sky filter is marred by excessive false positives. The other methods all perform well at 1-min data intervals; the method developed here provides more accurate detection at longer data averaging intervals. We show that the parameters are directly linked to the data frequency in the hope that these input variables may not have to be optimized for every data frequency and location. However, only a single fixed system in one location was carefully examined. Finally, we illustrate how accurate determination of clear-sky conditions helps to eliminate data noise and bias in the assessment of long-term performance of PV plants.

14 SOLAR ENERGY↗
DOE OSTI2024

Deep probabilistic direction prediction in 3D with applications to directional dark matter detectors

Abstract We present the first method to probabilistically predict 3D direction in a deep neural network model. The probabilistic predictions are modeled as a heteroscedastic von Mises-Fisher distribution on the sphere S 2 , giving a simple way to quantify aleatoric uncertainty. This approach generalizes the cosine distance loss which is a special case of our loss function when the uncertainty is assumed to be uniform across samples. We develop approximations required to make the likelihood function and gradient calculations stable. The method is applied to the task of predicting the 3D directions of electrons, the most complex signal in a class of experimental particle physics detectors designed to demonstrate the particle nature of dark matter and study solar neutrinos. Using simulated Monte Carlo data, the initial direction of recoiling electrons is inferred from their tortuous trajectories, as captured by the 3D detectors. For 40 keV electrons in a 70% He 30% CO 2 gas mixture at STP, the new approach achieves a mean cosine distance of 0.104 (26 ∘ ) compared to 0.556 (64 ∘ ) achieved by a non-machine learning algorithm. We show that the model is well-calibrated and accuracy can be increased further by removing samples with high predicted uncertainty. This advancement in probabilistic 3D directional learning could increase the sensitivity of directional dark matter detectors.

Computer Science↗
DOE OSTI2026

Intrinsic nonlocality of spin- and polarization-resolved probabilities in strong-field quantum electrodynamics

Spin and polarization are central to precision tests of fundamental physics and for interpreting radiation from astrophysical sources and ultraintense laser-matter experiments. Here, focusing on the fundamental process of nonlinear Compton scattering, we demonstrate that a key assumption underlying current strong-field quantum electrodynamics models, i.e., that emission can be treated as an instantaneous random event sampled from a local differential rate, is inconsistent once emission angles, electron spin, and/or photon polarization are resolved. Namely, even in strictly constant and uniform fields , the resulting fully differential distribution is sign indefinite, yielding negative inferred probabilities. The physical reason is that the photon emission probability builds up over a finite length of the electron trajectory, the formation region, during which the electron direction changes by roughly the same small angle that defines the radiation cone. Therefore, we put forward a new method where we integrate over this formation region analytically to obtain a physically consistent electron spin and photon polarization model. We show that the implementation of our model is compatible with existing Monte Carlo and particle-in-cell workflows. Simulations of a GeV-class electron-laser collision accessible at current petawatt facilities and of emission in a pulsarlike magnetic field are shown to reveal spin and polarization patterns that differ even qualitatively from state-of-the-art local models. In particular, our new model predicts substantial angle-dependent circular photon polarization where the well-known collinear-emission approach yields none, and a pronounced helicity bias in the recoiling electrons absent from current predictions. These findings have direct implications for upcoming strong-field QED experiments and for interpreting polarized radiation from extreme astrophysical environments.

astrophysical electromagnetic fields↗
NASA NTRSDate not supplied

Aerodynamic Implications of Aerocapture Systems for Uranus Orbiters

Exploration of the Ice Giants, Uranus in particular, has been indicated as the highest priority new flagship-class mission by the 2022 Planetary Science Decadal Survey. However, due to it’s location in the solar system, extensive travel time and resources are required for a mission to reach Uranus. Aerocapture offers shorter interplanetary transit times and reduced propellant requirements for such a mission when compared to a traditional propulsive-capture orbiter. Considerations for the aerodynamics of an aerocapture system enabling flagship-class science mission to Uranus are presented. The applicability of a low lift-to-drag entry vehicle aeroshell with flight heritage to a new planet and atmosphere is evaluated. A preliminary aerodatabase is formulated from low-fidelity methods and heritage flight mission data to provide initial estimates of trajectory space to be considered for inserting a spacecraft into orbit at Uranus using aerocapture. Computational fluid dynamics calculations in the free molecular, transitional, and continuum regimes using Direct Simulation Monte Carlo and Navier-Stokes solutions are used to assess the validity of this preliminary aerodatabase. The considered entry vehicle is found to varying levels of agreement with work done by previous flight missions when evaluated for flight in Uranus’ atmosphere. Implications for the aerodynamics of this vehicle as applied in an aerocapture system in flight at Uranus as found by the current work are discussed.

Eli R. Shellabarger↗
NASA NTRSDate not supplied

Aerodynamic Implications of Aerocapture Systems for Uranus Orbiters

Exploration of the Ice Giants, Uranus in particular, has been indicated as the highest priority new flagship-class mission by the 2022 Planetary Science Decadal Survey. However, due to it's location in the solar system, extensive travel time and resources are required for a mission to reach Uranus. Aerocapture offers shorter interplanetary transit times and reduced propellant requirements for such a mission when compared to a traditional propulsive-capture orbiter. Considerations for the aerodynamics of an aerocapture system enabling flagship-class science mission to Uranus are presented. The applicability of a low lift-to-drag entry vehicle aeroshell with flight heritage to a new planet and atmosphere is evaluated. A preliminary aerodatabase is formulated from low-fidelity methods and heritage flight mission data to provide initial estimates of trajectory space to be considered for inserting a spacecraft into orbit at Uranus using aerocapture. Computational fluid dynamics calculations in the free molecular, transitional, and continuum regimes using Direct Simulation Monte Carlo and Navier-Stokes solutions are used to assess the validity of this preliminary aerodatabase. The considered entry vehicle is found to varying levels of agreement with work done by previous flight missions when evaluated for flight in Uranus' atmosphere. Implications for the aerodynamics of this vehicle as applied in an aerocapture system in flight at Uranus as found by the current work are discussed.

Uranus↗
DOE OSTI2024

Neural simulation-based inference of the neutron star equation of state directly from telescope spectra

Neutron stars provide a unique opportunity to study strongly interacting matter under extreme density conditions. The intricacies of matter inside neutron stars and their equation of state are not directly visible, but determine bulk properties, such as mass and radius, which affect the star's thermal X-ray emissions. However, the telescope spectra of these emissions are also affected by the stellar distance, hydrogen column, and effective surface temperature, which are not always well-constrained. Uncertainties on these nuisance parameters must be accounted for when making a robust estimation of the equation of state. In this study, we develop a novel methodology that, for the first time, can infer the full posterior distribution of both the equation of state and nuisance parameters directly from telescope observations. This method relies on the use of neural likelihood estimation, in which normalizing flows use samples of simulated telescope data to learn the likelihood of the neutron star spectra as a function of these parameters, coupled with Hamiltonian Monte Carlo methods to efficiently sample from the corresponding posterior distribution. Our approach surpasses the accuracy of previous methods, improves the interpretability of the results by providing access to the full posterior distribution, and naturally scales to a growing number of neutron star observations expected in the coming years.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1992

Reflectance of broken cloud fields - Simulation and parameterization

The transfer of solar irradiance in plane parallel and broken cloud fields is simulated using a Monte Carlo method. The angular distribution pattern of radiances exiting the cloud layer is studied with varying cloud geometries, optical thicknesses, cloudiness, and solar zenith angles. A rather large anisotropy of the reflected flux is found, usually increasing with solar zenith angle and with patterns that strongly depend on cloud geometry. The main features are (1) a local maximum of reflected intensity in the forward direction for all cases, (2) a limb darkening for the plane parallel case, and (3) a limb brightening and a local maximum of reflected intensity in the backward direction for broken clouds. A parameterization for the azimuth-average reflectance function is developed. It reproduces the Monte Carlo simulation with a reasonable accuracy and illustrates that, when azimuthally averaged, the reflectance function is dominated by side viewing and intercloud shadowing effects.

Breon, Francois-Marie↗
DOE OSTI2020

A Dynamic Trajectory Fit to Multisensor Fireball Observations

Meteorites with known orbital origins are key to our understanding of solar system formation and the source of life on Earth. Fireball networks have been developed globally in a unified effort to record and ultimately retrieve these cosmic samples. However, the accuracy of the determined orbit and the likelihood of meteorite recovery depend directly on the accuracy of the chosen meteoroid triangulation method. There are three leading techniques for meteoroid triangulation discussed in the literature: the method of planes, the straight-line least-squares method, and the multiparameter fit method. Here we describe an alternative method to meteoroid triangulation, called the dynamic trajectory fit. This approach uses the meteoroid’s 3D dynamic equations of motion to fit a realistic trajectory directly to multisensor line-of-sight observations. This method has the ability to resolve fragmentation events, fit systematic observatory timing offsets, and determine mass estimates of the meteoroid along its observable trajectory. Through a comprehensive Monte Carlo analysis of over 100,000 trajectory simulations, we find this new method to more accurately estimate meteoroid trajectories of slow entry events (<25 km s{sup −1}) and events observed from low convergence angles (<10°) compared to existing meteoroid triangulation techniques. Additionally, we triangulate an observed fireball event with visible fragmentation using the various triangulation methods to show that the proposed dynamic trajectory fit implementing fragmentation to best match the captured multisensor line-of-sight data.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1993

Numerical simulation of rarefied flow through a slit. I - Direct simulation Monte Carlo results

The pressure-driven flow of a rarefied monatomic gas through a two-dimensional slit is simulated using the direct simulation Monte Carlo technique. Of particular interest is the change in flow field structure as pressure ratio and Knudsen number are varied. Comparisons are made to quantify the limits of validity of free-molecular theory and approximate, nearly free-molecular iterative methods. Also addressed is the sensitivity of the numerical solutions to grid structure and boundary conditions. The free-molecular theory is found to predict quantitative flow field properties (e.g., centerline velocities or downstream flux profiles) reasonably well for large finite Knudsen number with the error dependent on the pressure ratio. The nearly free-molecular corrections are shown to have limited range of applicability. A previously derived parameter is found to correlate total mass flux well as a function of pressure ratio and Knudsen number over a large portion of the transitional regime.

Wadsworth, D. C.↗
NASA NTRS1993

Zonally-decoupled DSMC solutions of hypersonic blunt body wake flows

Direct simulation Monte Carlo (DSMC) solutions are presented for the hypersonic flow behind a blunt body in which the wake region is solved in a zonally-decoupled manner. The forebody flow is solved separately using either a DSMC or a Navier-Stokes method, and the forebody exit plane solution is specified as the inflow condition to the decoupled DSMC solution of the wake region. Results are presented for a 70-deg, blunted cone at flow conditions that can be accommodated in existing low-density wind tunnels with the Knudsen number based on base diameter ranging from 0.03 to 0.001. The zonally-decoupled solutions show good agreement with fully-coupled DSMC solutions of the wake flow densities and velocities. The wake closure predicted by the zonally-decoupled solutions is in better agreement with fully-coupled results than that predicted by a fully-coupled Navier-Stokes method indicating the need to account for rarefaction in the wake for the cases considered. The combined use of Navier-Stokes for the forebody with a decoupled DSMC solution for the wake provides an efficient method for solving transitional blunt-body flows where the forebody flow is continuum and the wake is rarefied.

Wilmoth, Richard G.↗
NASA NTRS2016

Simulation of Satellite, Airborne and Terrestrial LiDAR with DART (I):Waveform Simulation with Quasi-Monte Carlo Ray Tracing

Light Detection And Ranging (LiDAR) provides unique data on the 3-D structure of atmosphere constituents and the Earth's surface. Simulating LiDAR returns for different laser technologies and Earth scenes is fundamental for evaluating and interpreting signal and noise in LiDAR data. Different types of models are capable of simulating LiDAR waveforms of Earth surfaces. Semi-empirical and geometric models can be imprecise because they rely on simplified simulations of Earth surfaces and light interaction mechanisms. On the other hand, Monte Carlo ray tracing (MCRT) models are potentially accurate but require long computational time. Here, we present a new LiDAR waveform simulation tool that is based on the introduction of a quasi-Monte Carlo ray tracing approach in the Discrete Anisotropic Radiative Transfer (DART) model. Two new approaches, the so-called "box method" and "Ray Carlo method", are implemented to provide robust and accurate simulations of LiDAR waveforms for any landscape, atmosphere and LiDAR sensor configuration (view direction, footprint size, pulse characteristics, etc.). The box method accelerates the selection of the scattering direction of a photon in the presence of scatterers with non-invertible phase function. The Ray Carlo method brings traditional ray-tracking into MCRT simulation, which makes computational time independent of LiDAR field of view (FOV) and reception solid angle. Both methods are fast enough for simulating multi-pulse acquisition. Sensitivity studies with various landscapes and atmosphere constituents are presented, and the simulated LiDAR signals compare favorably with their associated reflectance images and Laser Vegetation Imaging Sensor (LVIS) waveforms. The LiDAR module is fully integrated into DART, enabling more detailed simulations of LiDAR sensitivity to specific scene elements (e.g., atmospheric aerosols, leaf area, branches, or topography) and sensor configuration for airborne or satellite LiDAR sensors.

Gastellu-Etchegorry, Jean-Philippe↗
DOE OSTI2021

Dose Rate Analysis of the WCS Consolidated Interim Storage Facility

This report describes a confirmatory calculation for a proposed consolidated interim storage facility (CISF) to support the US Nuclear Regulatory Commission’s review of the license application submitted by the Interim Storage Partners. The scope of this report is limited to an assessment of the annual dose from direct radiation associated with Phase 1 of the Waste Control Specialists (WCS) CISF and the determination of the minimum controlled area boundary based on the 10 CFR 72.104 limit of 25 mrem annual dose to the whole body. The confirmatory dose rate calculations used the source term and shielding calculation capabilities of the SCALE 6.2.3 computer code system. The calculation method employed in the confirmatory calculations uses a detailed Monte Carlo radiation transport simulation from source to dose rate. This method differs from the two-step method used in the safety analysis report (SAR), which requires determination of the photon and neutron energy and angular distributions on the cask external surface and then the use of this surface source in a new radiation transport calculation to determine dose rate as a function of distance from the storage facility. A simulation was made using the complete site geometry (all casks present) but with only one cask containing source, and a total of 467 independent calculations were performed to obtain the dose rate maps produced by each storage cask. Basic input data and assumptions used in the SAR (e.g., cask design parameters and design basis assembly characteristics) were used in this confirmatory calculation for consistency with the SAR. Local atmospheric conditions for skyshine calculations were simulated using US standard atmosphere data, which describe average air temperature, pressure, and density as a function of altitude. The determined minimum required distances from the WCS Phase 1 CISF to a contour defined by the 25 mrem annual dose limit were approximately 623 m (0.387 miles) in the N-NE direction, 594 m (0.369 miles) in the SSW direction, and 533 m (0.331 miles) in the E-SE direction and the W-NW direction. The annual dose results of the confirmatory calculation are in relatively good agreement with the annual dose values presented in the WCS SAR. The annual dose produced by the Phase 1 CISF at the nearest site boundary, 0.75 miles away from the CISF, was estimated by this study to be approximately 6.02E-02 ± 2.0E-04 mrem, which is significantly smaller than the 25 mrem annual dose limit provided in 10 CFR 72.104. The SAR annual dose rate value for the nearest site boundary was 7.52E-2 mrem, which included 7.73E-03 mrem due to postulated leakage of the FO-, FC-, and FF-canisters. A comparison with the dose rate values reported in the SAR for 16 detector locations, D1 through D16, on the protected area boundary is also included in this report. This comparison shows that the SAR determined higher annual dose values for detectors D1 through D14 by 12% to 45% and lower annual dose values for detectors D15 and D16 by 18% and 8%, respectively, as compared with this ORNL confirmatory calculation. Sensitivity studies have been performed to support a comprehensive review of the application. In these studies, the effects on dose rate of the modeled air volume, density, and humidity were analyzed. The sensitivity calculations showed that the air volume used in the Monte Carlo simulations was adequate and that a larger air volume would not further increase the dose rate due to skyshine contributions. The sensitivity analysis showed that simulation of local atmospheric conditions is important for the accuracy of dose rate estimates at large distances from the facility. Air humidity was demonstrated to have negligible effects on dose rate values for the simulated local atmospheric conditions.

61 RADIATION PROTECTION AND DOSIMETRY↗
NASA NTRS1984

Production of high-energy gamma rays by cosmic ray interactions in the atmosphere and lunar surface

The production of gamma rays above 10 MeV by the interaction of cosmic rays in the atmosphere and the lunar surface is simulated using Monte Carlo methods. The calculation incorporates a new model of high-energy nucleon-nucleus interactions based on empirical fits to inclusive cross sections for the production of pions and nucleons. The atmospheric gamma ray flux is calculated as a function of direction, energy, and atmospheric depth. These calculations are compared with observations from balloons and from the SAS 2 satellite. Estimates of the flux of earth albedo electrons produced by cosmic ray interactions are presented. The lunar gamma ray albedo is calculated and compared with an upper limit based on SAS 2 measurements.

Morris, D. J.↗
NASA NTRS1991

On recontamination and directional-bias problems in Monte Carlo simulation of PDF turbulence models

Turbulent combustion can not be simulated adequately by conventional moment closure turbulence models. The difficulty lies in the fact that the reaction rate is in general an exponential function of the temperature, and the higher order correlations in the conventional moment closure models of the chemical source term can not be neglected, making the applications of such models impractical. The probability density function (pdf) method offers an attractive alternative: in a pdf model, the chemical source terms are closed and do not require additional models. A grid dependent Monte Carlo scheme was studied, since it is a logical alternative, wherein the number of computer operations increases only linearly with the increase of number of independent variables, as compared to the exponential increase in a conventional finite difference scheme. A new algorithm was devised that satisfies a restriction in the case of pure diffusion or uniform flow problems. Although for nonuniform flows absolute conservation seems impossible, the present scheme has reduced the error considerably.

Hsu, Andrew T.↗