Application of linear multifrequency-grey acceleration to preconditioned Krylov iterations for thermal radiation transport

Highlights

• Sn thermal radiation transport at high temperatures requires implicit timestepping.

• Linear-multi-frequency-grey (LMFG) acceleration is applied as a preconditioner.

• LMFG plus Krylov reduces time and required iterations for convergence.

• When scattering is added, formulations without inner iterations are superior.

Abstract

The thermal radiative transfer (TRT) equations comprise a radiation equation coupled to the material internal energy equation. Linearization of these equations produces effective, thermally-redistributed scattering through absorption-reemission. We investigate the effectiveness and efficiency of Linear-Multi-Frequency-Grey (LMFG) acceleration that has been reformulated for use as a preconditioner to Krylov iterative solution methods. We introduce two general frameworks, the scalar flux formulation (SFF) and the absorption rate formulation (ARF), and investigate their iterative properties in the absence and presence of true scattering. SFF has a group-dependent state size but may be formulated without inner iterations in the presence of scattering, while ARF has a group-independent state size but requires inner iterations when scattering is present. We compare and evaluate the computational cost and efficiency of LMFG applied to these two formulations using a direct solver for the preconditioners. This work is novel because the use of LMFG for the radiation transport equation, in conjunction with Krylov methods, involves special considerations not required for radiation diffusion.

Introduction

At high temperatures, thermal radiation (photons) born from blackbody emission dominates the internal energy in a system [1]. Even at more moderate temperatures, this radiation can be an important, nonlocal source of energy deposition. High-fidelity modeling uses a kinetic description of the radiation with a source term from the material, which is coupled to the radiation through photon absorption, emission, and scattering processes. At high material temperatures, absorption-reemission timescales can be orders of magnitude smaller than the desired timestep sizes — which resolve heating timescales — requiring the use of implicit time discretizations. The implicit material emission rate is nonlinear in the end-of-timestep material temperature. Linearization of emission about the material temperature is necessary for a linear method to have stability. It introduces within-timestep absorption-reemission term that depend implicitly on the end-of-timestep radiation distribution. The computation of that often requires an iterative solve of a stiff linear system. In this work, we explore a preconditioning technique, called linear multifrequency-grey (LMFG) acceleration, to improve convergence of the within-timestep iterative solution for the radiation distribution.

Formally, the radiation distribution is modeled with the time-, frequency-, angularly-, and spatially-dependent Boltzmann radiation transport equation. There are two common treatments for the dependence of the radiation distribution on photon frequency. The first treatment, referred to as the grey approximation, is to integrate the radiation equation over all frequencies and use approximate, frequency-averaged material interaction probabilities. The second treatment, referred to as the multigroup (MG) approximation, is to integrate the radiation equation over a collection of frequency groups and use material interaction probabilities averaged over each group, replacing the continuous-frequency radiation equation with a set of coupled, frequency-integrated equations called the MG radiation equations. A common simplifying assumption for the photon angular dependence, accurate for thick systems with large amounts of absorption-reemission, leads to a time-dependent radiation diffusion equation.

The grey or MG radiation diffusion equation can itself be used to calculate the radiation distribution, or it can be used to precondition the radiation transport. The grey radiation diffusion equation can also be used to precondition the MG radiation diffusion equation. LMFG is a linear preconditioner that accelerates the convergence of the MG radiation equations by solving a grey radiation diffusion equation. The LMFG acceleration technique has been proven to be effective for improving the iterative convergence of MG radiation diffusion calculations in high energy density simulations [2]. In [3], a grey radiation transport equation was used to accelerate the MG radiation transport equation. This method, called GTA, shifts the workload from MG radiation transport to grey radiation transport, resulting in improved computational efficiency.

The purpose of this paper is to investigate LMFG for MG thermal radiation transport calculations. Prior work has successfully demonstrated the effectiveness of LMFG to precondition the iterative solution of the MG radiation diffusion equation, especially when solved with Krylov method such as GMRES [4]. The transport setting is novel. In our MG transport setting, two distinct iterative formulations appear, one dealing with scalar fluxes and another dealing with absorption rate densities. We denote these methods as the scalar flux formulation (SFF) and the absorption rate formulation (ARF). In prior radiation diffusion work, only the ARF method was investigated. We investigate the iterative convergence properties of SFF and ARF in the absence and presence of physical scattering. As in [4], we formulate the methods as preconditioned linear systems that can be solved with Krylov iterative methods.

The remainder of this paper is organized as follows. In Section 2, we derive the discretized TRT equations, introduce the source-iteration scheme for iteratively solving these equations, and demonstrate the need for preconditioning. In Section 3, we introduce LMFG acceleration, SFF, and ARF for the transport equation, allowing for monoenergetic scattering. In Section 4, we investigate algorithms for dealing with the complications of this scattering. In Section 5, we present computational results comparing the performance of LMFG with Krylov methods to LMFG with source iteration; the performance of ARF to SFF; and the performance of various techniques for dealing with scattering. In Section 6, we give conclusions and recommendations for future work.