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A Parallel, Energy-Stable Low-Rank Integrator for Nonlinear Multi-Scale Thermal Radiative Transfer

Chinmay Patwardhan, Jonas Kusch

TL;DR

This work tackles the computational challenge of nonlinear thermal radiative transfer in multi-scale regimes by combining a macro--micro decomposition with dynamical low-rank approximation. It introduces a parallel, asymptotic-preserving, mass-conservative BUG-based integrator that evolves all low-rank factors in tandem and incorporates reflection-transmission boundary conditions efficiently. The full-rank macro--micro scheme is proven to be AP and energy-stable under a mixed hyperbolic-parabolic CFL condition, and its low-rank parallel counterpart inherits AP and energy stability through a conservative truncation strategy; boundary handling is carefully integrated. Numerical experiments on Marshak-wave and hohlraum problems demonstrate accurate diffusion-limit behavior, energy decay, and substantial computational savings with controlled rank growth, highlighting the method's potential for high-dimensional, nonlinear RT simulations.

Abstract

Thermal radiative transfer models physical phenomena ranging from supernovas in astrophysics to radiation from a hohlraum striking a fusion target in plasma physics. Transport and absorption of particles in radiative transfer at different rates lead to a complex interaction between the material and particles that involves highly varying time scales. Resolving these effects can require prohibitively small step sizes, which, combined with nonlinear effects and the particle density's high-dimensional phase space, render conventional numerical methods computationally expensive. This work presents an asymptotic--preserving, mass conservative, rank-adaptive, and parallel integrator for a macro--micro decomposition-based dynamical low-rank approximation of the thermal radiative transfer equations. The proposed integrator efficiently incorporates reflection-transmission type boundary conditions in the low-rank factors. It captures the nonlinear effects of thermal radiation and is energy stable with the step size restriction capturing both hyperbolic and parabolic CFL conditions. The efficacy of the proposed integrator is demonstrated with numerical experiments.

A Parallel, Energy-Stable Low-Rank Integrator for Nonlinear Multi-Scale Thermal Radiative Transfer

TL;DR

This work tackles the computational challenge of nonlinear thermal radiative transfer in multi-scale regimes by combining a macro--micro decomposition with dynamical low-rank approximation. It introduces a parallel, asymptotic-preserving, mass-conservative BUG-based integrator that evolves all low-rank factors in tandem and incorporates reflection-transmission boundary conditions efficiently. The full-rank macro--micro scheme is proven to be AP and energy-stable under a mixed hyperbolic-parabolic CFL condition, and its low-rank parallel counterpart inherits AP and energy stability through a conservative truncation strategy; boundary handling is carefully integrated. Numerical experiments on Marshak-wave and hohlraum problems demonstrate accurate diffusion-limit behavior, energy decay, and substantial computational savings with controlled rank growth, highlighting the method's potential for high-dimensional, nonlinear RT simulations.

Abstract

Thermal radiative transfer models physical phenomena ranging from supernovas in astrophysics to radiation from a hohlraum striking a fusion target in plasma physics. Transport and absorption of particles in radiative transfer at different rates lead to a complex interaction between the material and particles that involves highly varying time scales. Resolving these effects can require prohibitively small step sizes, which, combined with nonlinear effects and the particle density's high-dimensional phase space, render conventional numerical methods computationally expensive. This work presents an asymptotic--preserving, mass conservative, rank-adaptive, and parallel integrator for a macro--micro decomposition-based dynamical low-rank approximation of the thermal radiative transfer equations. The proposed integrator efficiently incorporates reflection-transmission type boundary conditions in the low-rank factors. It captures the nonlinear effects of thermal radiation and is energy stable with the step size restriction capturing both hyperbolic and parabolic CFL conditions. The efficacy of the proposed integrator is demonstrated with numerical experiments.

Paper Structure

This paper contains 21 sections, 15 theorems, 127 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Thermal radiative transfer equations
  4. Macro--micro decomposition
  5. Initial and boundary conditions
  6. Dynamical low-rank approximation
  7. An AP scheme for the macro--micro equations
  8. Angular discretisation
  9. Spatio-temporal discretization
  10. Local mass conservation
  11. AP property
  12. Energy stability
  13. An AP parallel low-rank scheme for macro--micro equations
  14. Boundary conditions
  15. AP property
  16. ...and 6 more sections

Key Result

Theorem 1

The full-rank macro--micro scheme eq:FDNMM is locally conservative for Dirichlet zero or periodic boundary conditions. That is, for $n \in\{0,1\}$, the scheme fulfills the discrete conservation law with an analogous definition of the conservation law at $(x_{{i}^{+}_{1/2}},y_{{j}^{+}_{1/2}})$.

Figures (7)

  • Figure 1: Two-dimensional staggered grid as described in MR3459981. $h$ and $T$ are evaluated at the red circles whereas $g$ is evaluated at the blue diamonds.
  • Figure 2: Left: Relative error of $T^{\mathrm{Full}}$ and $T^{\vartheta}$ compared to $T^{R}$ for $\varepsilon\in \{1,5\cdot 10^{-1}, 10^{-1},5\cdot 10^{-2},10^{-2},10^{-3},10^{-4}\}$. Right: Energy decay of the full solver and parallel BUG solver for $\varepsilon\in\{1.0,10^{-4}\}$.
  • Figure 3: Cross-section of the material and scalar flux for the Marshak wave test case at times $1$, $2$, $3$, $4$, and $5$ ps through $y=0.001$ for $\varepsilon=1.0$.
  • Figure 4: Cross-section of the material and scalar flux for the Marshak wave test case at time $5$ ps through $y=0.001$ for the diffusive limit with $\varepsilon=10^{-4}$.
  • Figure 5: Left: Geometry of the hohlraum as described in MR2657865. Right: Rank over time for the hohlraum test case with $\vartheta = 10^{-2}$ until $1~\mathrm{ns}$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Remark 1
  • Theorem 1: Local mass conservation
  • proof
  • Remark 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4: Local mass conservation
  • proof
  • ...and 21 more