Efficient SN-like and PN-like Dynamic Low Rank methods for Thermal Radiative Transfer

TL;DR

The paper tackles the high dimensionality of grey TRT by introducing two Dynamic Low Rank (DLR) schemes: a PN-like method based on an even-parity formulation and an SN-like method using collocation with time-evolving angular bases. The PN-like DLR achieves efficient, positive-definite solves via a pair of second-order systems, while the SN-like DLR leverages transport sweeps with collocation angles and DEIM to approximate the angular dependence with a low-rank representation. Numerical experiments in 2D show that both methods significantly reduce angular artifacts (ray effects) at costs comparable to traditional SN methods, with PN-like delivering smoother angular results and SN-like offering production-code compatibility and improved accuracy at modest ranks. Together, these approaches offer practical, scalable pathways to integrate dynamic low-rank TRT solvers into existing production codes, potentially enabling substantial savings in memory and compute time for complex multi-physics simulations.

Abstract

Dynamic Low Rank (DLR) methods are a promising way to reduce the computational cost and memory footprint of the high-dimensional thermal radiative transfer (TRT) equations. The TRT equations are a system of nonlinear PDEs that model the energy exhchange between the material temperature and the radiation energy density; due to their high dimensionality, solving the TRT equations is often bottleneck in multi-physics simulations. DLR methods represent the solution in terms of time-evolving SVD-like factors of angle and space. Although previous work has explored DLR methods for TRT, most of the methods have limitations that make them impractical for realistic scenarios and uncompetitive with current non-DLR production codes. Here we develop new PN-like and SN-like Dynamic Low Rank (DLR) methods for TRT. In the SN-like DLR method, we use the time-evolving angular basis functions to select time-evolving angles; this DLR formulation enables us to use the highly optimized SN transport sweep as our main computational kernel, and results in a practical way of leveraging low-rank methods in production TRT codes. In contrast, our PN-like DLR method uses an even-parity formulation and results in positive-definite linear systems to solve for each time step. We demonstrate the methods on several challenging, highly heterogenous problems in two spatial dimensions $(4$D) that these DLR schemes can give significant reduction in angular artifacts (``ray effects'') with the same cost as gold-standard SN methods.

Figures (4)

• Figure 1: Mesh and materials for the lattice problem defined in brunner2023family. The blue-green block in the center of the spatial domain corresponds to hot iron; the red blocks correspond to diamond; the yellow blocks correspond to cold iron; and the blue blocks correspond to foam.
• Figure 2: PN-like DLR method with rank $8$, SN-like DLR method with rank $16$, S6 ($18$ angles), and S10 ($50$ angles) for the lattice problem. Material temperature at times $t=3$, $t=14$, and $t=25$.
• Figure 3: Material regions, mesh, and initial conditions for the hohlraum problem.
• Figure 4: PN-like DLR method with rank $8$, SN-like DLR method with rank $16$, and S6 ($36$ angles) for the hohlraum problem. Plots show the material temperature at times $t=0.3$, $t=1.2$, and $t=3.0$.