Jeffery D. Densmore

A hybrid transport-diffusion method for Monte Carlo radiative-transfer simulations

Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Monte Carlo particle-transport simulations in diffusive media. If standard Monte Carlo is used in such media, particle histories will consist of many small steps, resulting in a computationally expensive calculation. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many small Monte Carlo steps, thus increasing the efficiency of the simulation. In addition, given that DDMC is based on a diffusion equation, it should produce accurate solutions if used judiciously. In practice, DDMC is combined with standard Monte Carlo to form a hybrid transport-diffusion method that can accurately simulate problems with both diffusive and non-diffusive regions. In this paper, we extend previously developed DDMC techniques in several ways that improve the accuracy and utility of DDMC for nonlinear, time-dependent, radiative-transfer calculations. The use of DDMC in these types of problems is advantageous since, due to the underlying linearizations, optically thick regions appear to be diffusive. First, we employ a diffusion equation that is discretized in space but is continuous in time. Not only is this methodology theoretically more accurate than temporally discretized DDMC techniques, but it also has the benefit that a particles time is always known. Thus, there is no ambiguity regarding what time to assign a particle that leaves an optically thick region (where DDMC is used) and begins transporting by standard Monte Carlo in an optically thin region. Also, we treat the interface between optically thick and optically thin regions with an improved method, based on the asymptotic diffusion-limit boundary condition, that can produce accurate results regardless of the angular distribution of the incident Monte Carlo particles. Finally, we develop a technique for estimating radiation momentum deposition during the DDMC simulation, a quantity that is required to calculate correct fluid motion in coupled radiation-hydrodynamics problems. With a set of numerical examples, we demonstrate that our improved DDMC method is accurate and can provide efficiency gains of several orders of magnitude over standard Monte Carlo.

A hybrid transport-diffusion Monte Carlo method for frequency-dependent radiative-transfer simulations

Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Implicit Monte Carlo radiative-transfer simulations in optically thick media. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many smaller Monte Carlo steps, thus improving the efficiency of the simulation. In this paper, we present an extension of DDMC for frequency-dependent radiative transfer. We base our new DDMC method on a frequency-integrated diffusion equation for frequencies below a specified threshold, as optical thickness is typically a decreasing function of frequency. Above this threshold we employ standard Monte Carlo, which results in a hybrid transport-diffusion scheme. With a set of frequency-dependent test problems, we confirm the accuracy and increased efficiency of our new DDMC method.

A Consistent, Moment-Based, Multiscale Solution Approach for Thermal Radiative Transfer Problems

We present an efficient numerical algorithm for solving the time-dependent grey thermal radiative transfer (TRT) equations. The algorithm utilizes the first two angular moments of the TRT equations (Quasi-diffusion (QD)) together with the material temperature equation to form a nonlinear low-order (LO) system. The LO system is solved via the Jacobian-free Newton-Krylov method. The combined high-order (HO) TRT and LO-QD system is used to bridge the diffusion and transport scales. In addition, a “consistency” term is introduced to make the truncation error in the LO system identical to the truncation error in the HO equation. The derivation of the consistency term is rather general; therefore, it can be extended to a variety of spatial and temporal discretizations.

An Efficient and Time Accurate, Moment-Based Scale-Bridging Algorithm for Thermal Radiative Transfer Problems

We present physics-based preconditioning and a time-stepping strategy for a moment-based scale-bridging algorithm applied to the thermal radiative transfer equation. Our goal is to obtain (asymptotically) second-order time accurate and consistent solutions without nonlinear iterations between the high-order (HO) transport equation and the low-order (LO) continuum system within a time step. Modified equation analysis shows that this can be achieved via a simple predictor-corrector time stepping that requires one inversion of the transport operator per time step. We propose a physics-based preconditioning based on a combination of the nonlinear elimination technique and Fleck--Cummings linearization. As a result, the LO system can be solved efficiently via a multigrid preconditioned Jacobian-free Newton--Krylov method. For a set of numerical test problems, the physics-based preconditioner reduces the number of GMRES iterations by a factor of 3

Newton's Method for the Computation of k-Eigenvalues in SN Transport Applications

\sim

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

4 as compared to a standard preconditioner for advection-di...

A Hybrid Transport-Diffusion Algorithm for Monte Carlo Radiation-Transport Simulations on Adaptive-Refinement Meshes in XY Geometry

Abstract Recently, Jacobian-Free Newton-Krylov (JFNK) methods have been used to solve the k-eigenvalue problem in diffusion and transport theories. We propose an improvement to Newton’s method (NM) for solving the k-eigenvalue problem in transport theory that avoids costly within-group iterations or iterations over energy groups. We present a formulation of the k-eigenvalue problem where a nonlinear function, whose roots are solutions of the k-eigenvalue problem, is written in terms of a generic fixed-point iteration (FPI). In this way any FPI that solves the k-eigenvalue problem can be accelerated using the Newton approach, including our improved formulation. Calculations with a one-dimensional multigroup SN transport implementation in MATLAB provide a proof of principle and show that convergence to the fundamental mode is feasible. Results generated using a three-dimensional Fortran implementation of several formulations of NM for the well-known Takeda and C5G7-MOX benchmark problems confirm the efficiency of NM for realistic k-eigenvalue calculations and highlight numerous advantages over traditional FPI.

Asymptotic analysis of the spatial discretization of radiation absorption and re-emission in Implicit Monte Carlo

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 Analysis of Angular APproximation Methods for Time-Dependent Radiation Transport

Abstract Discrete Diffusion Monte Carlo (DDMC) is a technique for increasing the efficiency of Monte Carlo simulations in diffusive media. If standard Monte Carlo is employed in such a regime, particle histories will consist of many small steps, a situation that results in a computationally inefficient calculation. In DDMC, particles take discrete steps between spatial cells according to a discretized diffusion equation. Each discrete step replaces many smaller Monte Carlo steps, thus increasing the efficiency of the simulation. In addition, because DDMC is based on the diffusion approximation, it should yield accurate solutions if used judiciously. In this paper, we present a new DDMC method for linear, steady-state radiation transport on adaptive-refinement meshes in two-dimensional Cartesian geometry. Adaptive-refinement meshes are characterized by local refinement such that a spatial cell may have multiple neighboring cells across each face. We specifically examine the cases of (a) a regular mesh structure without refinement, (b) a refined mesh structure where neighboring cells differ in refinement, and (c) a boundary mesh structure representing the interface between a diffusive region (where DDMC is used) and a nondiffusive region (where standard Monte Carlo is employed). With numerical examples, we demonstrate that our new DDMC technique is accurate and can provide efficiency gains of two orders of magnitude over standard Monte Carlo.

Manufactured Solutions in the Thick Diffusion Limit

We perform an asymptotic analysis of the spatial discretization of radiation absorption and re-emission in Implicit Monte Carlo (IMC), a Monte Carlo technique for simulating nonlinear radiative transfer. Specifically, we examine the approximation of absorption and re-emission by a spatially continuous artificial-scattering process and either a piecewise-constant or piecewise-linear emission source within each spatial cell. We consider three asymptotic scalings representing (i) a time step that resolves the mean-free time, (ii) a Courant limit on the time-step size, and (iii) a fixed time step that does not depend on any asymptotic scaling. For the piecewise-constant approximation, we show that only the third scaling results in a valid discretization of the proper diffusion equation, which implies that IMC may generate inaccurate solutions with optically large spatial cells if time steps are refined. However, we also demonstrate that, for a certain class of problems, the piecewise-linear approximation yields an appropriate discretized diffusion equation under all three scalings. We therefore expect IMC to produce accurate solutions for a wider range of time-step sizes when the piecewise-linear instead of piecewise-constant discretization is employed. We demonstrate the validity of our analysis with a set of numerical examples.