@article{coale_anistratov_2024, title={A Variable Eddington Factor Model for Thermal Radiative Transfer with Closure Based on Data-Driven Shape Function}, ISSN={["2332-4325"]}, DOI={10.1080/23324309.2024.2327992}, abstractNote={A new variable Eddington factor (VEF) model is presented for nonlinear problems of thermal radiative transfer (TRT). The VEF model is data-driven and acts on known (a-priori) radiation-diffusion solutions for material temperatures in the TRT problem. A linear auxiliary problem is constructed for the radiative transfer equation (RTE) whose emission source and opacities are evaluated at these known material temperatures. The solution to this RTE approximates the specific intensity distribution in phase-space and time. It is applied as a shape function to define the Eddington tensor for the presented VEF model. The shape function computed via the auxiliary RTE problem will capture some degree of transport effects within the TRT problem. The VEF moment equations closed with this approximate Eddington tensor will thus carry with them these captured transport effects. In this study, the temperature data comes from multigroup P1, P1/3, and flux-limited diffusion radiative transfer models. The proposed VEF model can be interpreted as a transport-corrected diffusion reduced-order model. Numerical results are presented on the Fleck-Cummings test problem which models a supersonic wavefront of radiation. The VEF model is shown to improve accuracy by 1–2 orders of magnitude compared to the considered radiation-diffusion model solutions to the TRT problem.}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Coale, Joseph M. and Anistratov, Dmitriy Y.}, year={2024}, month={Mar} }
 @article{coale_anistratov_2024, title={A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation}, volume={509}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2024.113044}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Coale, Joseph M. and Anistratov, Dmitriy Y.}, year={2024}, month={Jul} }
 @article{brantley_anistratov_urbatsch_2023, title={Foreword Selected papers from the 2021 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering ( M&C 2021)}, volume={197}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2022.2157619}, abstractNote={Mathematics and computational methods are of fundamental importance in the design and analysis of nuclear systems. The “Math & Comp” research community is actively engaged in advancing numerical algorithms and computational methods to simulate nuclear systems with increasingly higher fidelity. The American Nuclear Society (ANS) 2021 International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2021) continued the rich tradition of this biennial conference series, showcasing preeminent computational science research for nuclear applications. M&C 2021 was an M&C conference like none before. Much of the planning for the conference was conducted amid the uncertainty of the COVID-19 pandemic via e-mail and Zoom sessions in 2020 and 2021. The conference was originally scheduled for April 11–15, 2021, in Raleigh, North Carolina. The pandemic resulted in the delay of the conference to October 3–7, 2021, and the COVID-19 Delta surge led to the last-minute decision to go fully virtual for the first time in the history of the conference. A constant focus in planning the conference was a strong commitment to excellence in the technical program, a hallmark of this series of conferences. The papers in this special issue of Nuclear Science and Engineering represent selected highlights of the research presented at M&C 2021. The themes of M&C 2021 were mathematical and computational methods, numerical analysis, computer codes, and high-performance computer architectures for solving problems in nuclear science and engineering. The M&C 2021 technical program was based on 14 technical tracks:}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Brantley, Patrick S. and Anistratov, Dmitriy Y. and Urbatsch, Todd J.}, year={2023}, month={Feb}, pages={III-IV} }
 @article{whitman_palmer_anistratov_greaney_2023, title={Accelerated Deterministic Phonon Transport With Consistent Material Temperature and Intensities}, volume={145}, ISSN={["2832-8469"]}, DOI={10.1115/1.4056140}, abstractNote={Abstract
               We present a method for deterministically solving the frequency and temperature dependent phonon radiative transport (PRT) equation in the single-mode relaxation time (SMRT) approximation in the self-adjoint angular flux (SAAF) form. To handle the nonlinear coupling between the phonon intensities and the material temperature, we apply a linearization approach that is similar to one in thermal radiative transport. This procedure leads to the PRT equation with pseudo-scattering. The method presented includes acceleration of both the inner pseudo-scattering source iterations and outer temperature iteration with a gray diffusion synthetic acceleration (DSA) and Anderson acceleration, respectively. We use the finite-element method to discretize the PRT equation in space and the method of discrete ordinates (SN) for angular discretization. The proposed method is verified by a gray method of manufactured solutions problem and demonstrated on a problem using temperature and direction dependent multigroup data from lithium aluminate (LiAlO2). The iterative performance of the acceleration method in each test is then compared to the unaccelerated method.}, number={1}, journal={ASME JOURNAL OF HEAT AND MASS TRANSFER}, author={Whitman, Nicholas H. and Palmer, Todd S. and Anistratov, Dmitriy Y. and Greaney, P. Alex}, year={2023}, month={Jan} }
 @article{coale_anistratov_2023, title={Re duce d order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor}, volume={296}, ISSN={["1879-1352"]}, DOI={10.1016/j.jqsrt.2022.108458}, abstractNote={A new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.}, journal={JOURNAL OF QUANTITATIVE SPECTROSCOPY & RADIATIVE TRANSFER}, author={Coale, Joseph M. and Anistratov, Dmitriy Y.}, year={2023}, month={Feb} }
 @article{anistratov_2021, title={Nonlinear iterative projection methods with multigrid in photon frequency for thermal radiative transfer}, volume={444}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2021.110568}, abstractNote={This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y.}, year={2021}, month={Nov} }
 @article{ghassemi_anistratov_2020, title={An Approximation Method for Time-Dependent Problems in High Energy Density Thermal Radiative Transfer}, volume={49}, ISSN={["2332-4325"]}, DOI={10.1080/23324309.2019.1709082}, abstractNote={Abstract We analyze a computational method for solving multidimensional thermal radiative transfer (TRT) problems based on the radiative transfer equation (RTE) with approximate time evolution operator. The variation in the specific intensity of radiation over each time interval is approximated by an exponential function. This approximation reduces the RTE to an equation of steady-state form with time-dependent coefficients and a modified collision rate density term. This enables one to avoid storage of the high-dimensional solution from the previous time level. The numerical method for TRT problems applies the RTE with the time-dependent approximation as a high-order problem in multilevel system of low-order quasidiffusion (LOQD) equations. The change rate in the intensity is evaluated by the solution of the LOQD equations for the moments of the specific intensity. We study the accuracy of the approximate method in modeling the evolution of temperature and radiation waves in TRT problems.}, number={1}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Ghassemi, Pedram and Anistratov, Dmitriy Y.}, year={2020}, month={Jan}, pages={31–50} }
 @article{ghassemi_anistratov_2020, title={Multilevel quasidiffusion method with mixed-order time discretization for multigroup thermal radiative transfer problems}, volume={409}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2020.109315}, abstractNote={In this paper we present a numerical method for solving multigroup thermal radiative transfer (TRT) problems in 2D Cartesian geometry. It is based on the Quasidiffusion (aka Variable Eddington Factor) method and defined by the multilevel system of multigroup high-order radiative transfer (RT) equations and multigroup and grey low-order equations for moments of the intensity with the exact closures. We apply time integration schemes of different orders of accuracy to approximate the high-order and low-order equations. The first-order scheme is used for the high-order RT equations. The second-order scheme is applied to the low-order equations. This improves the accuracy of the TRT solution while using robust and relatively inexpensive scheme for the high-order RT equations. The solution of the low-order equations is non-monotonic because the hyperbolic low-order Quasidiffusion (QD) equations are discretized by the second-order scheme. To reduce non-monotonicity of the low-order solution we apply a monotonization procedure to the discretized time-dependent low-order equations based on the Limited-Trapezoidal method. Numerical results of the Fleck-Cummings test are presented to demonstrate performance of the developed mixed-order time integration scheme for the multilevel system of high-order and low-order QD equations. We use this TRT test problem to analyze the convergence in time of the mixed-order scheme.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Ghassemi, Pedram and Anistratov, Dmitriy Y.}, year={2020}, month={May} }
 @article{cornejo_anistratov_smith_2019, title={Iteration Methods with Multigrid in Energy for Eigenvalue Neutron Diffusion Problems}, volume={193}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2019.1573601}, abstractNote={Abstract In this paper we present nonlinear multilevel methods with multiple grids in energy for solving the k-eigenvalue problem for multigroup neutron diffusion equations. We develop multigrid-in-energy algorithms based on a nonlinear projection operator and several advanced prolongation operators. The evaluation of the eigenvalue is performed in the space with smallest dimensionality by solving the effective one-group diffusion problem. We consider two-dimensional Cartesian geometry. The multilevel methods are formulated in discrete form for the second-order finite volume discretization of the diffusion equation. The homogenization in energy is based on a spatially consistent discretization of the group diffusion equations on coarse grids in energy. We present numerical results of model reactor-physics problems with 44 energy groups. They demonstrate performance and main properties of the proposed iterative methods with multigrid in energy.}, number={8}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Cornejo, Luke R. and Anistratov, Dmitriy Y. and Smith, Kord}, year={2019}, pages={803–827} }
 @article{anistratov_warsa_2018, title={Discontinuous Finite Element Quasi-Diffusion Methods}, volume={191}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2018.1450013}, abstractNote={Abstract In this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and the LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. We also show that the method with independent discretization has the asymptotic diffusion limit.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Anistratov, Dmitriy Y. and Warsa, James S.}, year={2018}, pages={105–120} }
 @article{anistratov_2019, title={Stability analysis of a multilevel quasidiffusion method for thermal radiative transfer problems}, volume={376}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2018.09.034}, abstractNote={In this paper we analyze a multilevel quasidiffusion (QD) method for solving time-dependent multigroup nonlinear radiative transfer problems which describe interaction of photons with matter. The multilevel method is formulated by means of the high-order radiative transfer equation and a set of low-order moment equations. The fully implicit scheme is used to discretize equations in time. The stability analysis is applied to the method in semi-continuous and discretized forms. To perform Fourier analysis, the system of equations of the multilevel method is linearized about an equilibrium solution. The effects of discretization with respect to different independent variables are studied. The multilevel method is shown to be stable and fast converging. We also consider a version of the method in which time evolution in the radiative transfer equation is treated by means of the α-approximation. The Fleck–Cummings test problem is used to demonstrate performance of the multilevel QD method and study its iterative stability.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y.}, year={2019}, month={Jan}, pages={186–209} }
 @article{warsa_anistratov_2018, title={Two-Level Transport Methods with Independent Discretization}, volume={47}, ISSN={["2332-4325"]}, DOI={10.1080/23324309.2018.1497991}, abstractNote={Abstract In this paper, we present two-level nonlinear iteration methods for the transport equation in 1D slab geometry approximated by means of the linear discontinuous finite element method (LDFEM). We develop transport schemes based on the quasidiffusion (QD) method in which the low-order QD (LOQD) equations are discretized by the linear continuous finite element method (LCFEM). This requires a mapping of the LCFEM low-order solution to the LDFEM high-order solution to define the scattering term. Several mappings are proposed and analyzed. Another proposed transport discretization scheme is based on the step characteristics for the transport equation and LCFEM for the LOQD equations. We also develop new nonlinear synthetic acceleration (NSA) methods based on the LCFEM discretization of the QD equation. To gain iterative stability, the NSA algorithms are combined with the nonlinear Krylov acceleration method. We present numerical results that demonstrate performance and basic properties of the proposed discretization schemes and iterative solution methods.}, number={4-6}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Warsa, James S. and Anistratov, Dmitriy Y.}, year={2018}, pages={424–450} }
 @article{o'brien_mattingly_anistratov_2017, title={Sensitivity Analysis of Neutron Multiplicity Counting Statistics Using First-Order Perturbation Theory and Application to a Subcritical Plutonium Metal Benchmark}, volume={185}, ISSN={["1943-748X"]}, DOI={10.1080/00295639.2016.1272988}, abstractNote={Abstract It is frequently important to estimate the uncertainty and sensitivity of measured and computed detector responses in subcritical experiments and simulations. These uncertainties arise from the physical construction of the experiment, uncertainties in the transport parameters, and counting uncertainties. Perturbation theory enables sensitivity analysis (SA) and uncertainty quantification on integral quantities like detector responses. The aim of our work is to apply SA to the statistics of subcritical neutron multiplicity counting distributions. Current SA methods have only been applied to mean detector responses and the eigenvalue. For multiplicity counting experiments, knowledge of the higher-order counting moments and their uncertainties are essential for a complete SA. We apply perturbation theory to compute the sensitivity of neutron multiplicity counting moments to arbitrarily high order. Each moment is determined by solving an adjoint transport equation with a source term that is a function of the adjoint solutions for lower-order moments. This enables moments of arbitrarily high order to be sequentially determined, and it shows that each moment is sensitive to the uncertainties of all lower-order moments. To close our SA of the moments, we derive forward transport equations that are functions of the forward flux and lower-order moment adjoint fluxes. We verify our calculations for the first three moments by comparison with multiplicity counting measurements of a subcritical plutonium metal sphere. For the first three moments, the most influential parameters are ranked, and the validity of first-order perturbation theory is demonstrated by examining the series truncation error. This enables a detailed SA of subcritical multiplicity counting measurements of fissionable material based on transport theory.}, number={3}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={O'Brien, Sean and Mattingly, John and Anistratov, Dmitriy}, year={2017}, month={Mar}, pages={406–425} }
 @article{anistratov_cornejo_jones_2017, title={Stability analysis of nonlinear two-grid method for multigroup neutron diffusion problems}, volume={346}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2017.06.014}, abstractNote={We present theoretical analysis of a nonlinear acceleration method for solving multigroup neutron diffusion problems. This method is formulated with two energy grids that are defined by (i) fine-energy groups structure and (ii) coarse grid with just a single energy group. The coarse-grid equations are derived by averaging of the multigroup diffusion equations over energy. The method uses a nonlinear prolongation operator. We perform stability analysis of iteration algorithms for inhomogeneous (fixed-source) and eigenvalue neutron diffusion problems. To apply Fourier analysis the equations of the method are linearized about solutions of infinite-medium problems. The developed analysis enables us to predict convergence properties of this two-grid method in different types of problems. Numerical results of problems in 2D Cartesian geometry are presented to confirm theoretical predictions.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y. and Cornejo, Luke R. and Jones, Jesse P.}, year={2017}, month={Oct}, pages={278–294} }
 @article{cornejo_anistratov_2017, title={The multilevel quasidiffusion method with multigrid in energy for eigenvalue transport problems}, volume={101}, ISSN={0149-1970}, url={http://dx.doi.org/10.1016/J.PNUCENE.2017.05.014}, DOI={10.1016/J.PNUCENE.2017.05.014}, abstractNote={A multilevel iterative method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional geometry is developed. This method is based on a system of group low-order quasidiffusion (LOQD) equations defined on a sequence of coarsening energy grids. The spatial discretization of the LOQD equations uses compensation terms which make it consistent with a high-order transport scheme on a given spatial grid. Different multigrid algorithms are applied to solve the multilevel system of group LOQD equations on grids in energy. The eigenvalue is evaluated from the LOQD problem on a coarsest grid. To further improve the efficiency of iterative schemes hybrid multigrid algorithms are developed. The numerical results of tests with a large number groups are presented to demonstrate performance of the proposed iterative schemes.}, journal={Progress in Nuclear Energy}, publisher={Elsevier BV}, author={Cornejo, Luke R. and Anistratov, Dmitriy Y.}, year={2017}, month={Nov}, pages={401–408} }
 @article{cornejo_anistratov_2016, title={Nonlinear Diffusion Acceleration Method with Multigrid in Energy for k-Eigenvalue Neutron Transport Problems}, volume={184}, ISSN={["1943-748X"]}, DOI={10.13182/nse16-78}, abstractNote={Abstract We present a multilevel method for solving multigroup neutron transport k-eigenvalue problems in two-dimensional Cartesian geometry. It is based on the nonlinear diffusion acceleration (NDA) method. The multigroup low-order NDA (LONDA) equations are formulated on a sequence of energy grids. Various multigrid cycles are applied to solve the hierarchy of multigrid LONDA equations. The algorithms developed accelerate transport iterations and are effective in solving the multigroup NDA low-order equations. We present numerical results for model reactor-physics problems with a large number of groups to demonstrate the performance of different iterative schemes.}, number={4}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Cornejo, Luke R. and Anistratov, Dmitriy Y.}, year={2016}, month={Dec}, pages={514–526} }
 @article{anistratov_azmy_2015, title={Iterative stability analysis of spatial domain decomposition based on block Jacobi algorithm for the diamond-difference scheme}, volume={297}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2015.05.033}, abstractNote={We study convergence of the integral transport matrix method (ITMM) based on a parallel block Jacobi (PBJ) iterative strategy for solving particle transport problems. The ITMM is a spatial domain decomposition method proposed for massively parallel computations. A Fourier analysis of the PBJ-based iterations applied to SN diamond-difference equations in 1D slab and 2D Cartesian geometries is performed. It is carried out for infinite-medium problems with homogeneous material properties. To analyze the performance of the ITMM with the PBJ algorithm and evaluate its potential in scalability we consider a limiting case of one spatial cell per subdomain. The analysis shows that in such limit the spectral radius of the iteration method is one without regard to values of the scattering ratio and optical thickness of the spatial cells. This implies lack of convergence in infinite medium. Numerical results of finite-medium problems are presented. They demonstrate effects of finite size of spatial domain on the performance of the iteration algorithm as well as its asymptotic behavior when the extent of the spatial domain increases. These numerical experiments also show that for finite domains iterative convergence to a finite criterion is achievable in a multiple of the sum of number of cells in each dimension.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y. and Azmy, Yousry Y.}, year={2015}, month={Sep}, pages={462–479} }
 @article{anistratov_jones_2015, title={Space-Angle Homogenization of the Step Characteristic Scheme}, volume={44}, ISSN={["2332-4325"]}, DOI={10.1080/23324309.2015.1076848}, abstractNote={We present a new homogenized discretization scheme for solving k-eigenvalue neutron transport problems on coarse grids in space and angle. The developed scheme for 1D slab geometry is based on the step characteristic (SC) method. It is algebraically consistent with fine-mesh SC equations. We analyze the sensitivity of the homogenized transport scheme to perturbations in homogenized cross-sections and other coefficients of the scheme. The obtained results demonstrate that the proposed scheme is stable to small perturbations in its parameters.}, number={4-5}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Anistratov, Dmitriy Y. and Jones, Jesse P.}, year={2015}, pages={215–228} }
 @article{tamang_anistratov_2014, title={A Multilevel projective method for solving the space-time multigroup neutron kinetics equations coupled with the heat transfer equation}, volume={177}, DOI={10.13182/nse13-42}, abstractNote={Abstract We present a computational method for adequate and efficient coupling of the multigroup neutron transport equation with the precursor and heat transfer equations. It is based on the multilevel nonlinear quasi-diffusion (QD) method for solving the multigroup transport equation. The system of equations includes the time-dependent high-order transport equation and time-dependent multigroup and effective one-group low-order QD equations. We also apply the α-approximation for the time-dependent high-order transport equation. This approach enables one to avoid storing the angular flux from the previous time step. Numerical results for model transient problems are presented.}, number={1}, journal={Nuclear Science and Engineering}, author={Tamang, A. and Anistratov, D. Y.}, year={2014}, pages={1–18} }
 @article{wieselquist_anistratov_morel_2014, title={A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes}, volume={273}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2014.05.011}, abstractNote={We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Wieselquist, William A. and Anistratov, Dmitriy Y. and Morel, Jim E.}, year={2014}, month={Sep}, pages={343–357} }
 @article{stehle_anistratov_adams_2014, title={A hybrid transport-diffusion method for 2D transport problems with diffusive subdomains}, volume={270}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2014.03.056}, abstractNote={Abstract In this paper we present a computational method based on the Simple Corner Balance (SCB) scheme for solving 2D transport problems in diffusive media. It utilizes decomposition of spatial domain into transport and diffusive subregions. This methodology uses the low-order equations of the Second-Moment (SM) method for the first two angular moments of the transport solution. These low-order SM equations are solved globally. The high-order transport solution is computed only in transport subregions. The transport boundary conditions at interfaces with neighbouring diffusion subregions are formulated using asymptotic analysis of SCB. We apply the quasidiffusion (Eddington) tensor to evaluate transport effects in the problem domain and determine spatial ranges of diffusive subregions. Numerical results are presented. They demonstrate the accuracy of the developed methodology for the SCB scheme.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Stehle, Nicholas D. and Anistratov, Dmitriy Y. and Adams, Marvin L.}, year={2014}, month={Aug}, pages={325–344} }
 @article{anistratov_jones_2014, title={Spatial Homogenization of Transport Discretization Schemes}, volume={43}, ISSN={["2332-4325"]}, DOI={10.1080/00411450.2014.914040}, abstractNote={Neutron transport problems for whole reactor core calculations result in a very large amount of unknowns. Some difficulties related to dimensionality of this kind of problem can be resolved by solving a well-posed homogenized transport problem on a coarse grid. We apply homogenization methodology to develop discretization schemes for solving k-eigenvalue problems on coarse meshes in 1D slab geometry. The step characteristic (SC) method is used for fine-mesh transport calculations. We develop spatially homogenized transport schemes on a basis of two different transport discretization methods: SC and linear discontinuous schemes. The basic approach is to formulate a discretization that is spatially consistent with the given fine-mesh transport discretization. The presented numerical results demonstrate features of the developed methods in preserving grid functions of the angular flux computed with the SC method on fine spatial meshes. We study sensitivity of the proposed schemes to various types of perturbations in spatially averaged cross sections and other homogenization parameters.}, number={1-7}, journal={JOURNAL OF COMPUTATIONAL AND THEORETICAL TRANSPORT}, author={Anistratov, Dmitriy and Jones, Jesse}, year={2014}, pages={262–288} }
 @article{anistratov_2013, title={Multilevel NDA Methods for Solving Multigroup Eigenvalue Neutron Transport Problems}, volume={174}, ISSN={["0029-5639"]}, DOI={10.13182/nse12-28}, abstractNote={Abstract The nonlinear diffusion acceleration (NDA) method is an efficient and flexible transport iterative scheme for solving reactor-physics problems. This paper presents a fast iterative algorithm for solving multigroup neutron transport eigenvalue problems in one-dimensional slab geometry. The proposed method is defined by a multilevel system of equations that includes multigroup and effective one-group low-order NDA equations. The eigenvalue is evaluated in an exact projected solution space of the smallest dimensionality. Numerical results that illustrate the performance of the new algorithm are demonstrated.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Anistratov, Dmitriy Y.}, year={2013}, month={Jun}, pages={150–162} }
 @article{anistratov_stehle_2012, title={Computational transport methodology based on decomposition of a problem domain into transport and diffusive subdomains}, volume={231}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2012.06.008}, abstractNote={A large class of radiative transfer and particle transport problems contain highly diffusive regions. It is possible to reduce computational costs by solving a diffusion problem in diffusive subdomains instead of the transport equation. This enables one to decrease the dimensionality of the transport problem. In this paper we present a methodology for decomposition of a spatial domain of a transport problem into transport and diffusion subregions. We develop methods for solving one-group problems in 1D slab geometry. To identify and locate diffusive regions, we develop metrics for measuring transport effects that are based on the quasidiffusion (Eddington) factor. We present the results of test problems that demonstrate the accuracy of the proposed methodology.}, number={24}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, Dmitriy Y. and Stehle, Nicholas D.}, year={2012}, month={Oct}, pages={8009–8028} }
 @article{anistratov_gol'din_2011, title={Multilevel Quasidiffusion Methods for Solving Multigroup Neutron Transport k-Eigenvalue Problems in One-Dimensional Slab Geometry}, volume={169}, ISSN={["0029-5639"]}, DOI={10.13182/nse10-64}, abstractNote={Abstract The methods for solving k-eigenvalue problems for the multigroup neutron transport equation in one-dimensional slab geometry are presented. They are defined by means of multigroup and effective grey (one-group) low-order quasidiffusion (QD) equations. In this paper we formulate and study different variants of nonlinear QD iteration algorithms. These methods are analyzed on a set of test problems designed using C5G7 benchmark data. We present numerical results that demonstrate the performance of iteration schemes in different types of reactor physics problems. We consider tests that represent single-assembly and color-set calculations as well as a problem with elements of full-core computations involving a reflector zone.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Anistratov, Dmitriy Y. and Gol'din, Vladimir Ya}, year={2011}, month={Oct}, pages={111–132} }
 @article{roberts_anistratov_2010, title={Nonlinear Weighted Flux Methods for Particle Transport Problems in Two-Dimensional Cartesian Geometry}, volume={165}, ISSN={["0029-5639"]}, DOI={10.13182/nse08-48}, abstractNote={Abstract A family of nonlinear weighted flux (NWF) methods for solving the transport equation in two-dimensional (2-D) Cartesian geometry is considered. The low-order equations of these methods are defined by means of special linear-fractional factors that are determined by the high-order transport solution. An asymptotic diffusion limit analysis is performed on methods with a general weight function. The analysis revealed conditions on the weight necessary for an accurate approximation of the diffusion equation in this limit. We study methods with weights defined by linear and bilinear functions of directional cosines. As a result, we developed 2-D NWF methods formulated with the low-order equations that give rise to the diffusion equation in optically thick diffusive regions if their factors are calculated by means of the leading-order transport solution. The inherent asymptotic boundary conditions for the NWF methods are analyzed. Numerical results are presented to confirm theoretical results and demonstrate performance of the proposed methods.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Roberts, Loren and Anistratov, Dmitriy Y.}, year={2010}, month={Jun}, pages={133–148} }
 @article{constantinescu_anistratov_2009, title={STABILITY ANALYSIS OF THE QUASIDIFFUSION METHOD ON PERIODIC HETEROGENEOUS 1D TRANSPORT PROBLEMS}, volume={38}, ISSN={["1532-2424"]}, DOI={10.1080/00411450903372084}, abstractNote={We study the convergence of the quasidiffusion (QD) method on one-dimensional spatially periodic heterogeneous problems. The QD method is a nonlinear projection-iterative method. A Fourier analysis of the linearized QD equations is performed. The convergence rates of the QD method in the vicinity of the solution are obtained. We also analyze the Second Moment (SM) method, which can be interpreted as a linear version of the QD method. The presented analysis gives a new insight on the convergence behavior of the QD method in a discretized form and reveals the differences in the convergence of the QD and SM methods. Numerical results are presented to confirm theoretical predictions.}, number={6}, journal={TRANSPORT THEORY AND STATISTICAL PHYSICS}, author={Constantinescu, Adrian and Anistratov, Dmitriy Y.}, year={2009}, pages={295–316} }
 @article{roberts_anistratov_2007, title={Nonlinear weighted flux methods for particle transport problems}, volume={36}, ISSN={["1532-2424"]}, DOI={10.1080/00411450701703647}, abstractNote={A new parameterized family of iterative methods for the 1‐D slab geometry transport equation is proposed. The new methods are derived by integrating the transport equation over −1≤μ≤0 and 0≤μ≤1 with weight 1+β|μ|α, where α≥0. The asymptotic diffusion analysis enables us to determine a particular method of this family the solution of which satisfies a good approximation of both the diffusion equation and asymptotic boundary condition in the diffusive regions. Note that none of the α‐weighted nonlinear methods possesses this combination of properties. The convergence properties of the proposed method are similar to the properties of the diffusion‐synthetic acceleration (DSA), quasi‐diffusion, and DSA‐like α‐weighted nonlinear methods. Numerical results are presented to demonstrate the performance of the derived method.}, number={7}, journal={TRANSPORT THEORY AND STATISTICAL PHYSICS}, author={Roberts, L. and Anistratov, D. Y.}, year={2007}, pages={589–608} }
 @article{hiruta_anistratov_2006, title={Homogenization method for the two-dimensional low-order quasi-diffusion equations for reactor core calculations}, volume={154}, ISSN={["1943-748X"]}, DOI={10.13182/NSE06-A2637}, abstractNote={Abstract In this paper, we develop a homogenization methodology for the two-dimensional low-order quasi-diffusion equations for full-core reactor calculations that is based on a family of spatially consistent coarse-mesh discretization methods. The coarse-mesh solution generated by these methods preserves a number of spatial moments of the fine-mesh transport solution over each assembly. The proposed method reproduces accurately the complicated large-scale behavior of the transport solution within assemblies. To demonstrate the performance of the developed methodology, we present the numerical results of several test problems that simulate mixed-oxide-uranium and assembly-reflector interfacial effects.}, number={3}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hiruta, Hikaru and Anistratov, Dmitriy Y.}, year={2006}, month={Nov}, pages={328–352} }
 @article{anistratov_2005, title={Consistent spatial approximation of the low-order quasi-diffusion equations on coarse grids}, volume={149}, ISSN={["0029-5639"]}, DOI={10.13182/NSE05-A2485}, abstractNote={Abstract Spatial discretization methods have been developed for the low-order quasi-diffusion equations on coarse grids and corresponding homogenization procedure for full-core reactor calculations. The proposed methods reproduce accurately the complicated large-scale behavior of the transport solution within assemblies. The developed discretization is spatially consistent with a fine-mesh discretization of the transport equation in the sense that it preserves a set of spatial moments of the fine-mesh transport solution over either coarse-mesh cells or its subregions, as well as the surface currents and eigenvalue. To demonstrate accuracy of the proposed methods, numerical results are presented of calculations of test problems that simulate the interaction of mixed-oxide and uranium assemblies.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Anistratov, DY}, year={2005}, month={Feb}, pages={138–161} }
 @article{hiruta_anistratov_adams_2005, title={Splitting method for solving the coarse-mesh discretized low-order quasi-diffusion equations}, volume={149}, ISSN={["1943-748X"]}, DOI={10.13182/NSE05-A2486}, abstractNote={Abstract In this paper, the development is presented of a splitting method that can efficiently solve coarse-mesh discretized low-order quasi-diffusion (LOQD) equations. The LOQD problem can reproduce exactly the transport scalar flux and current. To solve the LOQD equations efficiently, a splitting method is proposed. The presented method splits the LOQD problem into two parts: (a) the D problem that captures a significant part of the transport solution in the central parts of assemblies and can be reduced to a diffusion-type equation and (b) the Q problem that accounts for the complicated behavior of the transport solution near assembly boundaries. Independent coarse-mesh discretizations are applied: the D problem equations are approximated by means of a finite element method, whereas the Q problem equations are discretized using a finite volume method. Numerical results demonstrate the efficiency of the methodology presented. This methodology can be used to modify existing diffusion codes for full-core calculations (which already solve a version of the D problem) to account for transport effects.}, number={2}, journal={NUCLEAR SCIENCE AND ENGINEERING}, author={Hiruta, H and Anistratov, DY and Adams, ML}, year={2005}, month={Feb}, pages={162–181} }
 @article{anistratov_larsen_2001, title={Nonlinear and linear alpha-weighted methods for particle transport problems}, volume={173}, ISSN={["1090-2716"]}, DOI={10.1006/jcph.2001.6905}, abstractNote={A parametrized family of iterative methods for the planar-geometry transport equation is proposed. This family is a generalization of previously proposed nonlinear flux methods. The new methods are derived by integrating the 1D transport equation over −1≤μ≤0 and 0≤μ≤1 with weight |μ|α, α≥0. Both nonlinear and linear methods are developed. The convergence properties of the proposed methods are studied theoretically by means of a Fourier stability analysis. The optimum value of α that provides the best convergence rate is derived. We also show that the convergence rates of nonlinear and linear methods are almost the same. Numerical results are presented to confirm these theoretical predictions.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Anistratov, DY and Larsen, EW}, year={2001}, month={Nov}, pages={664–684} }