Marvin L. Adams

Fast iterative methods for discrete-ordinates particle transport calculations

Abstract In discrete-ordinates (S N ) simulations of large problems involving linear interactions between radiation and matter, the underlying linear Boltzmann problem is discretized and the resulting system of algebraic equations is solved iteratively. If the physical system contains subregions that are optically thick with small absorption, the simplest iterative process, Source Iteration, is inefficient and costly. During the past 40 years, significant progress has been achieved in the development of acceleration methods that speed up the iterative convergence of these problems. This progress consists of ( i ) a theory to derive the acceleration strategies, ( ii ) a theory to predict the convergence properties of the new strategies, and ( iii ) the implementation of these concepts in production computer codes. In this Review we discuss the theoretical foundations of this work, the important results that have been accomplished, and remaining open questions.

Benchmark results for particle transport in a binary Markov statistical medium

Abstract We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.

Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems

Abstract The performance of discontinuous finite element methods (DFEMs) on problems that contain optically thick diffusive regions is analyzed and tested. The asymptotic analysis is quite general; it holds for an entire family of DFEMs in slab, XY, and XYZ geometries on arbitrarily connected polygonal or polyhedral spatial grids. The main contribution of the work is a theory that predicts and explains how DFEMs behave when applied to thick diffusive regions. It is well known that in the interior of such a region, the exact transport solution satisfies (to leading order) a diffusion equation, with boundary conditions that are known. Thus, in the interiors of such regions, the ideal discretized transport solution would satisfy (to leading order) an accurate discretization of the same diffusion equation and boundary conditions. The theory predicts that one class of DFEMs, which we call “zero-resolution” methods, fails dramatically in thick diffusive regions, yielding solutions that are completely meaningless. Another class—full-resolution methods—has leading-order solutions that satisfy discretizations of the correct diffusion equation. Full-resolution DFEMs are classified according to several categories of performance: continuity, robustness, accuracy, and boundary condition. Certain kinds of lumping, some of which are believed to be new, improve DFEM behavior in the continuity, robustness, and boundary-condition categories. Theoretical results are illustrated using different variations of linear and bilinear DFEMs on several test problems in XY geometry. In every case, numerical results agree precisely with the predictions of the asymptotic theory.

CRASH: A BLOCK-ADAPTIVE-MESH CODE FOR RADIATIVE SHOCK HYDRODYNAMICS-IMPLEMENTATION AND VERIFICATION

We describe the Center for Radiative Shock Hydrodynamics (CRASH) code, a block-adaptive-mesh code for multi-material radiation hydrodynamics. The implementation solves the radiation diffusion model with a gray or multi-group method and uses a flux-limited diffusion approximation to recover the free-streaming limit. Electrons and ions are allowed to have different temperatures and we include flux-limited electron heat conduction. The radiation hydrodynamic equations are solved in the Eulerian frame by means of a conservative finite-volume discretization in either one-, two-, or three-dimensional slab geometry or in two-dimensional cylindrical symmetry. An operator-split method is used to solve these equations in three substeps: (1) an explicit step of a shock-capturing hydrodynamic solver; (2) a linear advection of the radiation in frequency-logarithm space; and (3) an implicit solution of the stiff radiation diffusion, heat conduction, and energy exchange. We present a suite of verification test problems to demonstrate the accuracy and performance of the algorithms. The applications are for astrophysics and laboratory astrophysics. The CRASH code is an extension of the Block-Adaptive Tree Solarwind Roe Upwind Scheme (BATS-R-US) code with a new radiation transfer and heat conduction library and equation-of-state and multi-group opacity solvers. Both CRASH and BATS-R-US are part of the publicly available Space Weather Modeling Framework.

Diffusion Synthetic Acceleration of Discontinuous Finite Element Transport Iterations

The authors present a discretization of the diffusion equation that can be used to accelerate transport iterations when the transport equation is spatially differenced by a discontinuous finite element (DFE) method. That is, they present a prescription for diffusion synthetic acceleration of DFE transport iterations. (The well-known linear discontinuous and bilinear discontinuous schemes are examples of DFE transport differencings.) They demonstrate that the diffusion discretization can be obtained in any coordinate system on any grid. They show that the diffusion discretization is not strictly consistent with the transport discretization in the usual sense. Nevertheless, they find that it yields a scheme with unconditional stability and rapid convergence. Further, they find that as the optical thickness of spatial cells becomes large, the spectral radius of the iteration scheme approaches zero (i.e., instant convergence). They give analysis results for one- and two-dimensional Cartesian geometries and numerical results for one-dimensional Cartesian and spherical geometries.

Subcell balance methods for radiative transfer on arbitrary grids

Abstract We present a new spatial discretization method, which enforces conservation on quadrilateral subcells in an arbitrarily connected grid of polygonal cells, for two-dimensional radiative transfer problems. We review what is known about the performance of existing methods for optically thick, diffusive regions of radiative transfer problems, focusing in particular on bilinear discontinuous (BLD) finite-element methods and the simple corner-balance (SCB) method. We discuss the close relation of the SCB and BLD methods, and how they differ. By careful analysis, we relate specific properties of the SCB solution to specific approximations in the SCB method. We then build our new method by discarding those SCB approximations that lead to undesirable properties and carefully constructing new approximations designed to yield more desirable properties. We compare BLD, SCB. and the new scheme on a series of test problems in slab and XY geometries; numerical results invariably agree with predictions of the an...

Capturing the Effects of Unlike Neighbors in Single-Assembly Calculations

Abstract We present recent improvements in assembly-level calculations for reactor analysis, including modifications that support core-level analysis by quasi-diffusion. Our main focus is on accurately approximating the effects that neighboring assemblies have on the few-group cross sections, assembly discontinuity factors, form factors, and other transport parameters of a given assembly. We show that we can do this by using albedo boundary conditions that are estimated with low computational cost. We also present an efficient way to tabulate these effects to permit accurate interpolation by the core-level algorithm. We describe our algorithms and present results from several difficult test problems containing mixed-oxide and UO2 assemblies. Our methodology significantly reduces the largest errors made by present-day methodology. For example, in our test problems it reduces the maximum pin-power error by a factor of ˜5.

Transport synthetic acceleration for long-characteristics assembly-level transport problems

Abstract We apply the transport synthetic acceleration (TSA) scheme to the long-characteristics spatial discretization for the two-dimensional assembly-level transport problem. This synthetic method employs a simplified transport operator as its low-order approximation. Thus, in the acceleration step, we take advantage of features of the long-characteristics discretization that make it particularly well suited to assembly-level transport problems. Our main contribution is to address difficulties unique to the long-characteristics discretization and produce a computationally efficient acceleration scheme. The combination of the long-characteristics discretization, opposing reflecting boundary conditions (which are present in assembly-level transport problems), and TSA presents several challenges. We devise methods for overcoming each of them in a computationally efficient way. Since the boundary angular data exist on different grids in the high- and low-order problems, we define restriction and prolongation operations specific to the method of long characteristics to map between the two grids. We implement the conjugate gradient (CG) method in the presence of opposing reflection boundary conditions to solve the TSA low-order equations. The CG iteration may be applied only to symmetric positive definite (SPD) matrices; we prove that the long-characteristics discretization yields an SPD matrix. We present results of our acceleration scheme on a simple test problem, a typical pressurized water reactor assembly, and a typical boiling water reactor assembly.

A nonlinear corner-balance spatial discretization for transport on arbitrary grids

A strictly positive spatial discretization method for the linear transport equation is presented. This method, which is algebraically nonlinear, enforces particle conservation on subcells and approximates the spatial variation of the source in each subcell as an exponential. The method is described in slab geometry and analyzed in several limits of practical significance; numerical results are presented. An x-y-geometry version of the method is then presented, assuming a spatial grid of arbitrary polygons; numerical results are presented. A rapidly convergent method for accelerating the iterations on the scattering source is also presented and tested. The analyses and results demonstrate that the method is startlingly accurate, especially on shielding-type problems, even given coarse and/or distorted spatial meshes.

The asymptotic diffusion limit of a linear discontinuous discretization of a two-dimensional linear transport equation

Abstract Consider a linear transport problem, and let the mean free path and the absorption cross section be of size ϵ. It is well known that one obtains a diffusion problem as ϵ tends to zero. We discretize the transport problem on a fixed mesh, independent of ϵ, consider again the limit ϵ → 0 and ask whether one obtains an accurate discretization of the continuous diffusion problem. The answer is known to be affirmative for the linear discontinuous Galerkin finite element discretization in one space dimension. In this paper, we ask whether the same result holds in two space dimensions. We consider a linear discontinuous discretization based on rectangular meshes. Our main result is that the asymptotic limit of this discrete problem is not a discretization of the asymptotic limit of the continuous problem and thus that the discretization will be inaccurate in the asymptotic regime under consideration. We also propose a modified scheme which has the correct asymptotic behavior for spatially periodic problems, although not always for problems with boundaries. We present numerical results confirming our formal asymptotic analysis.