Nonlinear Iterative Projection Methods with Multigrid in Photon Frequency for Thermal Radiative Transfer

TL;DR

This work tackles the nonlinear thermal radiative transfer problem in 1D slab geometry, where a time-dependent, multigroup RT equation is tightly coupled to a nonlinear material energy balance. It introduces nonlinear projection methods that solve a high-order RT problem on a fine frequency grid together with low-order quasidiffusion (QD) equations on a hierarchy of frequency grids (MLQD), including grey low-order transport on the coarsest grid to update the material temperature. The method employs multigrid cycles (W, F, V) to move between frequency grids, with exact averaging of coefficients to form coarse-grid LOQD systems, and uses Newton’s method with a Fréchet derivative to handle opacity changes, yielding substantial reductions in LOQD solves and transport iterations. Numerical results on a Fleck-Cummings test with up to 256 frequency groups demonstrate rapid convergence and significant speedups, validating the approach’s scalability to very large group counts and suggesting extensions to scattering and more complex TRT physics through enhanced prolongation operators.

Abstract

This paper presents nonlinear iterative methods for the fundamental thermal radiative transfer (TRT) model defined by the time-dependent multifrequency radiative transfer (RT) equation and the material energy balance (MEB) equation. The iterative methods are based on the nonlinear projection approach and use multiple grids in photon frequency. They are formulated by the high-order RT equation on a given grid in photon frequency and low-order moment equations on a hierarchy of frequency grids. The material temperature is evaluated in the subspace of the lowest dimensionality from the MEB equation coupled to the effective grey low-order equations. The algorithms apply various multigrid cycles to visit frequency grids. Numerical results are presented to demonstrate convergence of the multigrid iterative algorithms in TRT problems with large number of photon frequency groups.

Figures (6)

• Figure 1: Diagrams of multigrid cycles on hierarchies of $\Gamma$ grids in frequency. E$_g^1$ - calculation of spectrum on the fine grid $\Omega_{\nu}^{1}$ by solving the multigroup LOQD equations, T - calculation of temperature by solving coupled grey LOQD and MEB equations on $\Omega_{\nu}^{\Gamma}$, E$_p^{\gamma}$ - calculation of spectrum on $\Omega_{\nu}^{\gamma}$ ($\gamma>1$) by solving the coarse-group LOQD equations.
• Figure 2: Numerical solution of the FC test with 256 groups and $\Delta t=2 \! \times \! 10^{-2}$ ns.
• Figure 3: Number of transport iterations ($M_{ti}$), cycles ($M_c$), and low-order solves ($M_{lo}$) at each time step in the FC test with 256 groups and $\Delta t = 2 \! \times \! 10^{-2}$ ns over $t\in$[0, 3 ns].
• Figure 4: Number of transport iterations ($M_{ti}$), cycles ($M_c$), and low-order solves ($M_{lo}$) at each time step in the FC test with 256 groups and $\Delta t = 4 \! \times \! 10^{-2}$ ns over $t\in$[0, 3 ns].
• Figure 5: Convergence of temperature ($||\Delta T^{(s)}||_{\infty}$ [eV]) over transport iterations at $t=8 \! \times \! 10^{-2}$ ns.