A Comparison of the Consistent and Independent Second Moment Methods Applied to Thermal Radiative Transfer

TL;DR

This work extends local discontinuous Galerkin Second Moment Methods (SMM) to time-dependent, multigroup thermal radiative transfer by coupling a gray diffusion low-order system to the energy balance via closures, enabling efficient Newton-type solves for stiff absorption-emission. It compares consistent and independent low-order discretizations, showing consistent gains higher accuracy on under-resolved grids, while independent can offer faster convergence on well-resolved grids; both leverage SPD diffusion operators for scalable preconditioned solves. Through 1D and 2D gray and multigroup benchmarks (Marshak, Larsen, Crooked Pipe, Brunner), the study demonstrates that piecewise-linear gray opacities are essential, and that SMM accelerates TRT by up to an order of magnitude or more compared to unaccelerated schemes, with trade-offs in robustness and solution quality depending on mesh resolution. The results provide guidance on discretization choices and solver strategies for practical TRT simulations requiring efficient resolution of stiff radiation-material coupling. Overall, the gray SMM approach offers a practical, scalable framework for accelerating TRT computations in high-energy-density and astrophysical contexts.

Abstract

The design of efficient numerical methods for modeling thermal radiative transfer (TRT) is challenging due to the stiff, nonlinear coupling between radiation and material energies, especially at the time scales of interest in high energy density physics and astrophysics. Here, we investigate the use of the Second Moment Method (SMM) to accelerate absorption-emission within the context of the multigroup, Discrete Ordinates transport equations with discontinuous Galerkin spatial discretization. SMM employs a reduced-dimensional, diffusion-based model of radiation transport that, when coupled with suitable discrete closures, serves as a proxy for the transport equation, isolating the transport equation from the stiff absorption-emission physics. We use a gray low-order system to reduce the cost of solving the low-order system and leverage SMM low-order discretizations specifically designed to be scalably solvable with existing linear solver technology. Our algorithm robustly resolves the nonlinear TRT system while only relying on transport sweeps, linearly solving symmetric and positive definite, gray diffusion systems, and nonlinearly solving the spatially pointwise energy balance equation. This algorithm is used as a vehicle to compare the efficacy of low-order discretizations developed for steady-state, linear transport on gray and multigroup TRT problems in one and two spatial dimensions.

Figures (23)

• Figure 1: A comparison of the error with respect to a spatially and temporally resolved reference solution when the SMMs use the low and high-order solution as the previous time step's solution.
• Figure 2: The (a) error and (b) efficiency as the mesh size, $h$, and time step size, $\Delta t$, are reduced. In (a), reference first-order lines in space and time are provided. In (b), efficiency is plotted as accuracy versus runtime. Each continuous line represents the efficiency associated with a fixed spatial mesh as the time step is reduced. The line labels indicate the number of mesh refinements with $r=2$, $r=3$, and $r=4$ representing meshes with 128, 256, and 512 elements, respectively.
• Figure 3: The (a) temperature, (b) energy density, and (c) flux on the coarse Marshak wave problem with 32 elements in space and a time step of 4e-3. A spatially and temporally resolved reference solution is included.
• Figure 4: The (a) temperature, (b) energy density, and (c) flux on the refined Marshak wave problem with 512 elements in space and a time step of 5e-4. A spatially and temporally resolved reference solution is included.
• Figure 5: Materials and boundary conditions for Larsen's problem.

Theorems & Definitions (2)

• Remark 1
• Remark 2