Dmitriy Y. Anistratov

Rylan C. Paye, Dmitriy Y. Anistratov, Jim E. Morel et al.

In this paper, new implicit methods with reduced memory are developed for solving the time-dependent Boltzmann transport equation (BTE). One-group transport problems in 1D slab geometry are considered. The reduced-memory methods are formulated for the BTE discretized with the linear-discontinuous scheme in space and backward-Euler time integration method. Numerical results are presented to demonstrate performance of the proposed numerical methods.

Vincent N. Novellino, Dmitriy Y. Anistratov

This paper presents multilevel hybrid transport (MLHT) methods for solving the neutral-particle Boltzmann transport equation. The proposed MLHT methods are formulated on a sequence of spatial grids using a multilevel Monte Carlo (MLMC) approach. The general MLMC algorithm is defined by recursively estimating the expected value of the correction to a solution functional on a neighboring grid. MLMC theory optimizes the total computational cost for estimating a functional to within a target accuracy. The proposed MLHT algorithms are based on the quasidiffusion (variable Eddington factor) and second-moment methods. For these methods, the low-order equations for the angular moments of the angular flux are discretized in space. Monte Carlo techniques compute the closures for the low-order equations; then the equations are solved, yielding a single realization of the global flux solution. The ensemble average of the realizations yields the level solution. The results for 1-D slab transport problems demonstrate weak convergence of the functionals. We observe that the variance of the correction factors decreases faster than the computational cost of generating an MLMC sample increases. In the problems considered, the variance and cost of the MLMC solution are driven by the coarse-grid calculations.

Joseph M. Coale, Dmitriy Y. Anistratov

This paper presents a new technique for developing reduced-order models (ROMs) for nonlinear radiative transfer problems in high-energy density physics. The proper orthogonal decomposition (POD) of photon intensities is applied to obtain global basis functions for the Galerkin projection (POD-Galerkin) of the time-dependent multigroup Boltzmann transport equation (BTE) for photons. The POD-Galerkin solution of the BTE is used to determine the quasidiffusion (Eddington) factors that yield closures for the nonlinear system of (i) multilevel low-order quasidiffusion (VEF) equations and (ii) material energy balance equation. Numerical results are presented to demonstrate accuracy of the ROMs obtained with different low-rank approximations of intensities.

Joseph M. Coale, Dmitriy Y. Anistratov

In this paper analysis is performed on a computational method for thermal radiative transfer (TRT) problems based on the multilevel quasidiffusion (variable Eddington factor) method with the method of long characteristics (ray tracing) for the Boltzmann transport equation (BTE). The method is formulated with a multilevel set of moment equations of the BTE which are coupled to the material energy balance (MEB). The moment equations are exactly closed via the Eddington tensor defined by the BTE solution. Two discrete spatial meshes are defined: a material grid on which the MEB and low-order moment equations are discretized, and a grid of characteristics for solving the BTE. Numerical testing of the method is completed on the well-known Fleck-Cummings test problem which models a supersonic radiation wave propagation. Mesh refinement studies are performed on each of the two spatial grids independently, holding one mesh width constant while refining the other. We also present the data on convergence of iterations.

Dmitriy Y. Anistratov, Joseph M. Coale

This paper presents approximation methods for time-dependent thermal radiative transfer problems in high energy density physics. It is based on the multilevel quasidiffusion method defined by the high-order radiative transfer equation (RTE) and the low-order quasidiffusion (aka VEF) equations for the moments of the specific intensity. A large part of data storage in TRT problems between time steps is determined by the dimensionality of grid functions of the radiation intensity. The approximate implicit methods with reduced memory for the time-dependent Boltzmann equation are applied to the high-order RTE, discretized in time with the backward Euler (BE) scheme. The high-dimensional intensity from the previous time level in the BE scheme is approximated by means of the low-rank proper orthogonal decomposition (POD). Another version of the presented method applies the POD to the remainder term of P2 expansion of the intensity. The accuracy of the solution of the approximate implicit methods depends of the rank of the POD. The proposed methods enable one to reduce storage requirements in time dependent problems. Numerical results of a Fleck-Cummings TRT test problem are presented.

Dmitriy Y. Anistratov, Joseph M. Coale, James S. Warsa et al.

This paper presents multilevel iterative schemes for solving the multigroup Boltzmann transport equations (BTEs) with parallel calculation of group equations. They are formulated with multigroup and grey low-order equations of the Second-Moment (SM) method. The group high-order BTEs and low-order SM (LOSM) equations are solved in parallel. To further improve convergence and increase computational efficiency of algorithms Anderson acceleration is applied to inner iterations for solving the system of multigroup LOSM equations. Numerical results are presented to demonstrate performance of the multilevel iterative methods.

Caleb A. Shaw, Dmitriy Y. Anistratov

This paper presents a new Monte Carlo (MC) algorithm for time-dependent particle transport problems with global variance reduction based on automatic weight windows (WWs). The centers of WWs at a time step are defined by the solution of an auxiliary hybrid MC / deterministic problem formed by the low-order second-moment (LOSM) equations. The closures for the hybrid LOSM equations are calculated by the MC method. The LOSM equations are discretized by a scheme of the second-order accuracy in time and space. Filtering techniques are applied to reduce noise effects in the LOSM closures. The WWs defined with the auxiliary solution give rise to sufficiently uniform MC particle distribution in space on each time step. The algorithm is analyzed by means of an analytic transport benchmark. We study performance of the MC algorithm depending on a set parameters of WWs. Figure of merit and relative error results are presented, demonstrating the performance of the hybrid MC method and quantifying its computational efficiency.

Joseph M. Coale, Dmitriy Y. Anistratov

In this paper we present a multilevel projection-based iterative scheme for solving thermal radiative transfer problems that performs iteration cycles on the high-order Boltzmann transport equation (BTE) and low-order moment equations. Fully implicit temporal discretization based on the backward Euler time-integration method is used for all equations. The multilevel iterative scheme is designed to perform iteration cycles over collections of multiple time steps, each of which can be interpreted as a coarse time interval with a subgrid of time steps. This treatment is demonstrated to transform implicit temporal integrators to diagonally-implicit multi-step schemes on the coarse time grid formed with the amalgamated time intervals. A multilevel set of moment equations are formulated by the nonlinear projective approach. The Eddington tensor defined with the BTE solution provides exact closure for the moment equations. During each iteration, a number of chronological time steps are solved with the BTE alone, after which the same collection of time steps is solved with the moment equations and material energy balance. Numerical results are presented to demonstrate the effectiveness of this iterative scheme for simulating evolving radiation and heat waves in 2D geometry.

Dmitriy Y. Anistratov

This paper presents an iteration method for solving linear particle transport problems in binary stochastic mixtures. It is based on nonlinear projection approach. The method is defined by a hierarchy of equations consisting of the high-order transport equation for materials, low-order Yvon-Mertens equations for conditional ensemble average of the material partial scalar fluxes, and low-order quasidiffusion equations for the ensemble average of the scalar flux and current. The multilevel system of equations is solved by means of an iterative algorithm with the $V$-cycle. The iteration method is analyzed on a set of numerical test problems.