Moment-based Acceleration Research Articles

Accelerating the solution of the [formula omitted] equations with highly anisotropic scattering using the Fokker-Planck approximation

The discrete ordinates method can model forward-peaked transport problems accurately. However, convergence of discrete ordinates solution can become arbitrarily slow upon use of standard iterative procedures like source iteration. Standard zeroth and first moment-based acceleration methods like nonlinear diffusion acceleration and diffusion synthetic acceleration are ineffective in accelerating such problems because these methods do not correct higher order Legendre-moments of angular flux. In this paper, we explore the idea of using the Fokker-Planck approximation as a preconditioner to accelerate the convergence of solution of forward-peaked transport problems. We observe that such preconditioning can successfully reduce the iteration count by up to four orders of magnitude, and solver runtime by up to three orders of magnitude when compared to diffusion synthetic acceleration.

Toward Asymptotic Diffusion Limit Preserving High-Order, Low-Order Method

Recent development of the high-order, low-order (HOLO) method has shown promising results for solving thermal radiative transfer problems. The HOLO algorithm is a moment-based acceleration, similar to the well-known nonlinear diffusion acceleration and coarse-mesh finite difference methods. In this work, we introduce a new spatial-differencing scheme for the low-order (LO) system based on the corner-balance method and analyze an asymptotic diffusion property for a one-dimensional gray equation. An asymptotic analysis indicates that the new spatial-differencing scheme possesses the equilibrium diffusion limit. Numerical examples demonstrate significant improvements in the solution accuracy compared to the LO finite-volume discretization with a discontinuous source reconstruction.

A Stable 1D Multigroup High-Order Low-Order Method

ABSTRACTThe high-order low-order (HOLO) method is a recently developed moment-based acceleration scheme for solving time-dependent thermal radiative transfer problems, and has been shown to exhibit orders of magnitude speedups over traditional time-stepping schemes. However, a linear stability analysis by Haut et al. (2015) revealed that the current formulation of the multigroup HOLO method was unstable in certain parameter regions. Since then, we have replaced the intensity-weighted opacity in the first angular moment equation of the low-order (LO) system with the Rosseland opacity. This results in a modified HOLO method (HOLO-R) that is significantly more stable.This paper has two primary components. First, we present results from a linear stability analysis of the HOLO-R method for a 2-group problem. These results show that HOLO-R is stable in all regions of the parameter space we considered, and strongly suggest that HOLO-R is unconditionally stable. The predicted decay factors from the stability analysis are verified with numerically computed decay factors from a spectral HOLO-R code. Second, we describe a preliminary (unoptimized) implementation of HOLO-R in Capsaicin – a deterministic, discrete-ordinates radiation transport code at Los Alamos National Laboratory. The results from this implementation further verify the stability of the HOLO-R method. Moreover, in the simulation of a 50-group, nonlinear Marshak wave problem, away from the 2-group, linearized stability analysis, we assess HOLO-R’s accuracy, efficiency, and robustness as compared to current methods in Capsaicin. To our knowledge, this is the first implementation of HOLO that uses a mass-lumped discontinuous Galerkin spatial discretization for both the high-order (HO) and LO systems, and details of this discretization are provided in the Appendix.

Applying Nonlinear Diffusion Acceleration to the Neutron Transport k-Eigenvalue Problem with Anisotropic Scattering

High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Furthermore, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

Monte Carlo simulation methods in moment-based scale-bridging algorithms for thermal radiative-transfer problems

We present a moment-based acceleration algorithm applied to Monte Carlo simulation of thermal radiative-transfer problems. Our acceleration algorithm employs a continuum system of moments to accelerate convergence of stiff absorption–emission physics. The combination of energy-conserving tallies and the use of an asymptotic approximation in optically thick regions remedy the difficulties of local energy conservation and mitigation of statistical noise in such regions. We demonstrate the efficiency and accuracy of the developed method. We also compare directly to the standard linearization-based method of Fleck and Cummings [1]. A factor of 40 reduction in total computational time is achieved with the new algorithm for an equivalent (or more accurate) solution as compared with the Fleck–Cummings algorithm.

Moment-Based Acceleration of Monte Carlo Solution for Multifrequency Thermal Radiative Transfer Problems

We have extended a Monte Carlo-based, moment-based acceleration algorithm to the solution of multifrequency thermal radiative transfer problems. This study focuses on two aspects. First, we consider stability/accuracy issues for a predictor-corrector time-stepping. It is demonstrated that with a consistent Planckian-weighted opacity the predictor-corrector algorithm can run stably with a larger time-step size compared to a fixed (or lagged) opacity case. With this advancement, consistency improves by about two orders of magnitude, while additional computational cost is kept minimal (< 10%).We also extend the “asymptotic assistance” concept to multifrequency problems. This technique replaces the multifrequency Monte Carlo solution with an asymptotic solution of the O(ε) accurate solution of the equilibrium diffusion limit in optically thick regions. With this technique, the computational time is reduced about a factor of four in a one-dimensional problem. Furthermore, the solution with asymptotic assistance can become more accurate for a similar computational time because it enables a finer group structure in the high-order problem.

Charge-and-energy conserving moment-based accelerator for a multi-species Vlasov–Fokker–Planck–Ampère system, part II: Collisional aspects

In this study, we extend the moment-based acceleration algorithm for the charge, momentum, and energy conserving Vlasov–Ampère discretization developed in Ref. [1] by including a reduced Fokker–Planck operator. We propose an energy conserving discretization for the Fokker–Planck collision operator. We show by numerical experiment that the new algorithm 1) efficiently converges the nonlinearly coupled Vlasov–Fokker–Planck–Ampère system, and 2) accurately steps over stiff time-scales such as the inverse electron plasma frequency, and the electron–electron collision time-scale. We demonstrate that discrete energy conservation is critical to eliminate numerical heating issues when strong density gradients exist.

Efficient Multiphysics Coupling for Fast Burst Reactors in Slab Geometry

An efficient neutron transport-based multiphysics coupling algorithm is developed to model fast-burst reactor dynamics. It extends the previously developed tightly coupled implicit-explicit (IMEX) multiphysics algorithm based on neutron diffusion and linear mechanics to follow the dynamic time scale of the problem. The diffusion model is not expected to adequately represent the neutronic behavior of the system due its small size. The transport effects are incorporated using the moment-based acceleration concept. This concept enables us to isolate the angular flux from the coupled multiphysics system by using a consistently discretized lower-order system for the scalar flux in the coupling equations, thus helping to mitigate the increased numerical burden associated with a neutron transport model. Solution of a representative slab model demonstrates the difference between diffusion and transport models and also demonstrates second-order convergence in the coupled simulation with the combination of the moment-based acceleration concept and the IMEX algorithm.

Moment-Based Acceleration for Neutral Gas Kinetics with BGK Collision Operator

In this work, we present a moment-based accelerator algorithm for a Picard iteration applied to a neutral gas dynamics Boltzmann transport equation with a Bhatnagar-Gross-Krook collision operator. Traditional approaches relying on either explicit or Picard iteration schemes (i.e., source iteration) are severely limited for investigating time-scales much larger than the collisional relaxation time-scale, τ. We have developed a nonlinear accelerator algorithm that allows one to step over this stiff collision time scale and follow the hydrodynamic time scale of the problem when appropriate. The new algorithm relies on formulating a nonlinear, coupled system comprised of a high-order (HO) kinetic equation and a low-order (LO) fluid moment equation system. The HO equation provides self-consistent closures to the LO fluid equations, while the latter provides the required implicit-moment variables to evaluate the collision operator. We characterize the performance of the new algorithm on a Sod shock tube and a strong shock tube problem with varying Knudsen number.

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

Leveraging Anderson Acceleration for improved convergence of iterative solutions to transport systems

In this note we demonstrate that using Anderson Acceleration (AA) in place of a standard Picard iteration can not only increase the convergence rate but also make the iteration more robust for two transport applications. We also compare the convergence acceleration provided by AA to that provided by moment-based acceleration methods. Additionally, we demonstrate that those two acceleration methods can be used together in a nested fashion. We begin by describing the AA algorithm. At this point, we will describe two application problems, one from neutronics and one from plasma physics, on which we will apply AA. We provide computational results which highlight the benefits of using AA, namely that we can compute solutions using fewer function evaluations, larger time-steps, and achieve a more robust iteration.