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Temporal-Stability-Enhanced and Energy-Stable Dynamical Low-Rank Approximation for Multiscale Linear Kinetic Transport Equations

Shun Li, Yan Jiang, Mengping Zhang, Tao Xiong

TL;DR

The paper addresses the computational challenge of linear kinetic transport equations across regimes by introducing an asymptotic-preserving dynamical low-rank method. The approach combines a macro-micro decomposition, a Schur-complement reduction, and an energy-consistent $S_N$ discretization to achieve unconditional stability in the diffusion limit and significant reductions in memory and compute. The authors prove energy stability for the full-rank scheme and extend it to a low-rank version using a carefully structured ansatz and a BUG/aBUG integrator, preserving the correct diffusion limit. Numerical results in 1D and 2D validate the AP property, energy dissipation, and substantial speedups over full-rank methods, including a lattice test with checkerboard regions.

Abstract

In this paper, we develop an asymptotic-preserving dynamical low-rank method for the multiscale linear kinetic transport equation. The proposed scheme is unconditionally stable in the diffusive regime while preserving the correct asymptotic behavior, and can achieve significant reductions in computational cost through a low-rank representation and large time step stability. A low-rank formulation consistent with the discrete energy is introduced under the discrete ordinates discretization, and energy stability of the resulting scheme is established. Numerical experiments confirm the energy stability and demonstrate that the method is efficient while maintaining accuracy across different regimes and capturing the correct asymptotic limits.

Temporal-Stability-Enhanced and Energy-Stable Dynamical Low-Rank Approximation for Multiscale Linear Kinetic Transport Equations

TL;DR

The paper addresses the computational challenge of linear kinetic transport equations across regimes by introducing an asymptotic-preserving dynamical low-rank method. The approach combines a macro-micro decomposition, a Schur-complement reduction, and an energy-consistent discretization to achieve unconditional stability in the diffusion limit and significant reductions in memory and compute. The authors prove energy stability for the full-rank scheme and extend it to a low-rank version using a carefully structured ansatz and a BUG/aBUG integrator, preserving the correct diffusion limit. Numerical results in 1D and 2D validate the AP property, energy dissipation, and substantial speedups over full-rank methods, including a lattice test with checkerboard regions.

Abstract

In this paper, we develop an asymptotic-preserving dynamical low-rank method for the multiscale linear kinetic transport equation. The proposed scheme is unconditionally stable in the diffusive regime while preserving the correct asymptotic behavior, and can achieve significant reductions in computational cost through a low-rank representation and large time step stability. A low-rank formulation consistent with the discrete energy is introduced under the discrete ordinates discretization, and energy stability of the resulting scheme is established. Numerical experiments confirm the energy stability and demonstrate that the method is efficient while maintaining accuracy across different regimes and capturing the correct asymptotic limits.
Paper Structure (19 sections, 7 theorems, 104 equations, 8 figures, 1 table)

This paper contains 19 sections, 7 theorems, 104 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Kinetic transport equation
  4. Dynamic Low-Rank Approximation
  5. Full-rank Method
  6. Temporal Discretization
  7. Angular and Spatial Discretization
  8. Energy stability of the full-rank method
  9. Low-Rank method
  10. Energy-Consistent Low-Rank Approximation
  11. Asymptotic-Preserving Property
  12. Zero Density Constraint for the Microscopic Component
  13. Energy Stability of the Energy-Consistent Low-Rank method
  14. Numerical results
  15. Gaussian initial data in one dimension
  16. ...and 4 more sections

Key Result

Theorem 3.1

Assume that $\Phi = 0$. The IMEX scheme eq:IMEX-2d is $1$-stable under the time step condition where $C_A = 1$, $C_B^{(j)} = \frac{1}{|D_{\bm{\Omega}}|} \bm{w}^\top \lvert*\rvert{\bm{\Omega}^{(j)}} \bm{1} = \frac{1}{2}$ for both one- and two-dimensional cases.

Figures (8)

  • Figure 3.1: Two-dimensional staggered grid einkemmer2021asymptoticpreserving. Red: $\rho$; Blue: $\bm{g}$.
  • Figure 5.1: Gaussian initial data in one dimension. Columns from left to right: $(\varepsilon, T) = (1, 1.0)$, $(10^{-2}, 0.2)$, and $(10^{-6}, 0.2)$. Rows from top to bottom: density profile, energy evolution, and rank of the microscopic component $\bm{g}$.
  • Figure 5.2: Energy evolution for various low-rank methods. The inset reveals that the unweighted IMEX-BUG scheme violates the energy dissipation principle via a non-physical growth.
  • Figure 5.3: $L^2$ error (top) and average wall time per step (bottom) for $\varepsilon = 1, 10^{-2}, 10^{-6}$ (left to right).
  • Figure 5.4: Gaussian initial data in two dimensions ($\varepsilon = 10^{-6}$). Density slice of solutions along $y = 0$ (left) at time $T = 0.1$, energy dissipation (middle), and rank of the microscopic component $\bm{g}$ of IMEX-S-BUG and IMEX-S-aBUG method (right).
  • ...and 3 more figures

Theorems & Definitions (15)

  • Definition 3.1
  • Theorem 3.1: Energy stability of the IMEX scheme
  • Theorem 3.2: Energy $\theta$-stability of the IMEX-S scheme
  • Corollary 3.3: Energy stability of the IMEX-S scheme
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 5 more