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Asymptotic-preserving and energy stable dynamical low-rank approximation for thermal radiative transfer equations

Chinmay Patwardhan, Martin Frank, Jonas Kusch

TL;DR

This work tackles the numerical difficulty of simulating gray thermal radiative transfer in slab geometry by combining a macro-micro decomposition with dynamical low-rank approximation. The authors develop an asymptotic-preserving and energy-stable scheme based on a macro-micro formulation and a modified augmented BUG integrator, ensuring correct diffusion behavior as the system approaches the Rosseland limit and enabling rank adaptation to balance accuracy and cost. They prove energy stability under a CFL condition that accounts for both kinetic and diffusive regimes and demonstrate the method’s efficacy through rectangular-pulse and absorber test cases, where low-rank solutions closely follow full moment (Pn) results while dissipating energy as expected. The proposed framework offers a scalable, structure-preserving approach for high-dimensional radiative transfer problems with multi-scale dynamics, with potential impact on simulations requiring reliable diffusion limits and reduced computational complexity.

Abstract

The thermal radiative transfer equations model temperature evolution through a background medium as a result of radiation. When a large number of particles are absorbed in a short time scale, the dynamics tend to a non-linear diffusion-type equation called the Rosseland approximation. The main challenges for constructing numerical schemes that exhibit the correct limiting behavior are posed by the solution's high-dimensional phase space and multi-scale effects. In this work, we propose an asymptotic-preserving and rank-adaptive dynamical low-rank approximation scheme based on the macro-micro decomposition of the particle density and a modified augmented basis-update \& Galerkin integrator. We show that this scheme, for linear particle emission by the material, dissipates energy over time under a step size restriction that captures the hyperbolic and parabolic CFL conditions. We demonstrate the efficacy of the proposed method in a series of numerical experiments.

Asymptotic-preserving and energy stable dynamical low-rank approximation for thermal radiative transfer equations

TL;DR

This work tackles the numerical difficulty of simulating gray thermal radiative transfer in slab geometry by combining a macro-micro decomposition with dynamical low-rank approximation. The authors develop an asymptotic-preserving and energy-stable scheme based on a macro-micro formulation and a modified augmented BUG integrator, ensuring correct diffusion behavior as the system approaches the Rosseland limit and enabling rank adaptation to balance accuracy and cost. They prove energy stability under a CFL condition that accounts for both kinetic and diffusive regimes and demonstrate the method’s efficacy through rectangular-pulse and absorber test cases, where low-rank solutions closely follow full moment (Pn) results while dissipating energy as expected. The proposed framework offers a scalable, structure-preserving approach for high-dimensional radiative transfer problems with multi-scale dynamics, with potential impact on simulations requiring reliable diffusion limits and reduced computational complexity.

Abstract

The thermal radiative transfer equations model temperature evolution through a background medium as a result of radiation. When a large number of particles are absorbed in a short time scale, the dynamics tend to a non-linear diffusion-type equation called the Rosseland approximation. The main challenges for constructing numerical schemes that exhibit the correct limiting behavior are posed by the solution's high-dimensional phase space and multi-scale effects. In this work, we propose an asymptotic-preserving and rank-adaptive dynamical low-rank approximation scheme based on the macro-micro decomposition of the particle density and a modified augmented basis-update \& Galerkin integrator. We show that this scheme, for linear particle emission by the material, dissipates energy over time under a step size restriction that captures the hyperbolic and parabolic CFL conditions. We demonstrate the efficacy of the proposed method in a series of numerical experiments.
Paper Structure (23 sections, 13 theorems, 136 equations, 5 figures)

This paper contains 23 sections, 13 theorems, 136 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Thermal radiative transfer equations
  4. Macro-micro decomposition
  5. Initial and boundary conditions
  6. Angular discretization of microscopic variable
  7. Dynamical low-rank approximation
  8. Energy stability of modal macro-micro equations
  9. Spatio-temporal discretization
  10. Stability analysis
  11. Dynamical low-rank approximation for the modal macro-micro equations
  12. Fixed-rank modal macro-micro BUG integrator
  13. Spatio-temporal discretization
  14. Energy stability
  15. Local mass conservation
  16. ...and 8 more sections

Key Result

Theorem 1

In the limit $\varepsilon \to 0$, the modal macro-micro scheme eq:FDLMM gives a consistent discretization of the diffusion equation

Figures (5)

  • Figure 1: Relative mass error for the rectangular pulse test case in the kinetic and diffusive regime.
  • Figure 2: Numerical results of the rectangular pulse test case in the kinetic regime, i.e., $\varepsilon = 1$ at $t = 1.5$. In the first row, we present the temperature profile at end-time for the moment and low-rank methods; in the second row, we have the corresponding scalar flux. In the last row, we have the energy of the system over time for all the methods and the rank evolution of the BUG integrator.
  • Figure 3: Numerical results of the rectangular pulse test case in the diffusive regime, i.e. $\varepsilon = 10^{-5}$ at $t = 1.5$. Top left: Temperature profile, Top right: Scalar flux, Bottom left: Energy of the system over time for all the methods, Bottom right: Rank evolution of rank-adaptive integrator over time.
  • Figure 4: Numerical results of the absorber test case in the kinetic regime, i.e., $\varepsilon = 1$ at $t = 1.5$. In the first row, we present the temperature profile at end-time for the moment and low-rank methods; in the second row, we have the corresponding scalar flux. In the last row, we have the energy of the system over time for all the methods and the rank evolution of the BUG integrator.
  • Figure 5: Numerical results of the absorber test case in the diffusive regime, i.e., $\varepsilon = 10^{-5}$ at $t = 1.5$. Top left: Temperature profile, Top right: Scalar flux, Bottom left: Energy of the system over time for all the methods, Bottom right: Rank evolution of rank-adaptive integrator over time.

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Theorem 2
  • Remark 4
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 17 more