One-sweep moment-based semi-implicit-explicit integration for gray thermal radiation transport

TL;DR

The paper addresses stiff gray TRT in multi-physics contexts by introducing a nonlinear moment-based partition (HOLO) and a semi-implicit-explicit Runge-Kutta (SIMEX-RK) time integration that enables one transport sweep per stage. By explicitly handling the emission source and opacities in the high-order part and solving a low-order nonlinear system implicitly, the method achieves stability across streaming and diffusion limits while reducing computational cost compared to fully implicit schemes. Demonstrations on 1D Marshak and 2D crooked pipe benchmarks show orders-of-magnitude speedups and higher accuracy with higher-order SIMEX-RK schemes, validating the approach for tabular opacities and potential coupling to hydrodynamics. The method promises practical impact by enabling efficient, accurate TRT within large-scale multiphysics simulations and is extensible to other moment formulations and future physics including scattering and multifrequency TRT.

Abstract

Thermal radiation transport (TRT) is a time dependent, high dimensional partial integro-differential equation. In practical applications such as inertial confinement fusion, TRT is coupled to other physics such as hydrodynamics, plasmas, etc., and the timescales one is interested in capturing are often much slower than the radiation timescale. As a result, TRT is treated implicitly, and due to its stiffness and high dimensionality, is often a dominant computational cost in multiphysics simulations. Here we develop a new approach for implicit-explicit (IMEX) integration of gray TRT in the deterministic S$_N$ setting, which requires only one sweep per stage, with the simplest first-order method requiring only one sweep per time step. The partitioning of equations is done via a moment-based high-order low-order formulation of TRT, where the streaming operator and first two moments are used to capture the asymptotic stiff regimes of the streaming limit and diffusion limit. Absorption-reemission is treated explicitly, and although stiff, is sufficiently damped by the implicit solve that we achieve stable accurate time integration without incorporating the coupling of the high order and low order equations implicitly. Due to nonlinear coupling of the high-order and low-order equations through temperature-dependent opacities, to facilitate IMEX partitioning and higher-order methods, we use a semi-implicit integration approach amenable to nonlinear partitions. Results are demonstrated on thick Marshak and crooked pipe benchmark problems, demonstrating orders of magnitude improvement in accuracy and wallclock compared with the standard first-order implicit integration typically used.

Figures (14)

• Figure 1: Location of HO and LO variables.
• Figure 2: Evolution of the temperature on the Marshak wave problem over time. Radiation impinges on the slab from the left. As the initially cold material heats, it becomes transparent, allowing radiation to penetrate further into the domain. The steep temperature gradient at the front of the wave poses significant challenges for robustness. Note that $1\sh = 10ns$.
• Figure 3: Convergence to a reference solution for the radiation energy density as a function of the time step size, $\Delta t$, for (a) implicit, (b) LIMEX-Euler, and (c) H-LDIRK2(2,2,2). A range of algorithmic choices are compared corresponding to the use of full or half range boundary conditions and the maximum, minimum, and harmonic average value for the interface opacity. Gray dashed lines denote first- and second-order convergence. Missing data points indicate a method was not stable at that time step size.
• Figure 4: The final temperature profile at $t=10\ns$. In (a), the interface opacity options are varied with the boundary condition type fixed to half range while (b) fixes the interface opacity type to maximum and varies the boundary condition type. All solutions used a moderate time step of size 8e-3. Implicit HOLO with a time step of size 3e-5 is used as the reference solution.
• Figure 5: Error with respect to the reference solution as the time step is decreased in the (a) radiation energy density and (b) temperature for a range of HOLO schemes. The time-resolved solution generated by SSP-LDIRK3(3,3,2) is used as the reference. Gray dashed lines denote first- and second-order convergence. Missing data points indicate a method was not stable at that time step size.

Theorems & Definitions (1)

• Remark 3.1: Nonlinearly partitioned RK methods