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Iterative methods of linearized moment equations for rarefied gases

Xiaoyu Dong, Zhenning Cai

TL;DR

This work develops and analyzes a family of iterative solvers for steady-state linearized moment equations derived from the Boltzmann equation. It starts with a block symmetric Gauss-Seidel (BSGS) method for the discretized moment system and shows slower convergence as the Knudsen number $\epsilon$ decreases; to counter this, it couples BSGS with micro-macro decomposition (BSGS-MM) and extends to multiscale decompositions (BSGS-MS) and hybrids to cover a broad range of $\epsilon$. The paper provides Fourier-based convergence analyses for the first-order BSGS, the BSSR variant, and the BSGS-MM method and corroborates the theory with one- and two-dimensional numerical experiments, including heat transfer and cavity flows. The results indicate that micro-macro and multiscale decompositions substantially improve convergence in dense regimes, while plain BSGS remains effective for highly rarefied flows; hybridizations offer robust performance across regimes. These methods offer scalable, accurate solvers for large moment systems and lay groundwork for extending to nonlinear moment closures.

Abstract

We study the iterative methods for large moment systems derived from the linearized Boltzmann equation. By Fourier analysis, it is shown that the direct application of the block symmetric Gauss-Seidel (BSGS) method has slower convergence for smaller Knudsen numbers. Better convergence rates for dense flows are then achieved by coupling the BSGS method with the micro-macro decomposition, which treats the moment equations as a coupled system with a microscopic part and a macroscopic part. Since the macroscopic part contains only a small number of equations, it can be solved accurately during the iteration with a relatively small computational cost, which accelerates the overall iteration. The method is further generalized to the multiscale decomposition which splits the moment system into many subsystems with different orders of magnitude. Both one- and two-dimensional numerical tests are carried out to examine the performances of these methods. Possible issues regarding the efficiency and convergence are discussed in the conclusion.

Iterative methods of linearized moment equations for rarefied gases

TL;DR

This work develops and analyzes a family of iterative solvers for steady-state linearized moment equations derived from the Boltzmann equation. It starts with a block symmetric Gauss-Seidel (BSGS) method for the discretized moment system and shows slower convergence as the Knudsen number decreases; to counter this, it couples BSGS with micro-macro decomposition (BSGS-MM) and extends to multiscale decompositions (BSGS-MS) and hybrids to cover a broad range of . The paper provides Fourier-based convergence analyses for the first-order BSGS, the BSSR variant, and the BSGS-MM method and corroborates the theory with one- and two-dimensional numerical experiments, including heat transfer and cavity flows. The results indicate that micro-macro and multiscale decompositions substantially improve convergence in dense regimes, while plain BSGS remains effective for highly rarefied flows; hybridizations offer robust performance across regimes. These methods offer scalable, accurate solvers for large moment systems and lay groundwork for extending to nonlinear moment closures.

Abstract

We study the iterative methods for large moment systems derived from the linearized Boltzmann equation. By Fourier analysis, it is shown that the direct application of the block symmetric Gauss-Seidel (BSGS) method has slower convergence for smaller Knudsen numbers. Better convergence rates for dense flows are then achieved by coupling the BSGS method with the micro-macro decomposition, which treats the moment equations as a coupled system with a microscopic part and a macroscopic part. Since the macroscopic part contains only a small number of equations, it can be solved accurately during the iteration with a relatively small computational cost, which accelerates the overall iteration. The method is further generalized to the multiscale decomposition which splits the moment system into many subsystems with different orders of magnitude. Both one- and two-dimensional numerical tests are carried out to examine the performances of these methods. Possible issues regarding the efficiency and convergence are discussed in the conclusion.
Paper Structure (24 sections, 2 theorems, 89 equations, 17 figures, 3 tables)

This paper contains 24 sections, 2 theorems, 89 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Upwind finite volume method for the linearized Boltzmann equation
  3. Linearized Boltzmann equation and moment equations
  4. Spatial discretization
  5. Block symmetric Gauss-Seidel method to solve the linearized Boltzmann equation
  6. Block symmetric Gauss-Seidel method for the first-order scheme
  7. Block symmetric successive relaxation method for the second-order scheme
  8. Coupling with the micro-macro decomposition
  9. BSGS method with micro-macro decomposition
  10. Variations of the BSGS-MM method
  11. The Hybrid BSGS-MM method
  12. The multiscale BSGS (BSGS-MS) method
  13. The Hybrid BSGS-MS method
  14. Numerical examples
  15. One-dimensional tests: heat transfer
  16. ...and 9 more sections

Key Result

Theorem 1

\newlabelthm:lambda0 Let $\eta \in (0,2\pi)$ and $\Delta x > 0$. Assume that $\lambda_0$, $\lambda_1$ and $\boldsymbol{w}_0(v)$, $\boldsymbol{w}_1(v)$, $\boldsymbol{w}_2(v)$ satisfy eq:eps_order for any $v \in \mathbb{R}$, and $\boldsymbol{w}_0(v)$ is a nonzero function. Then it holds that $|\lamb

Figures (17)

  • Figure 1: The distributions of density and temperature with $N=5$.
  • Figure 1: The distribution of the maximum value of $|\lambda_0|$.
  • Figure 1: The distribution of the maximum value of $|\lambda|$.
  • Figure 2: The errors of upwind finite volume schemes for various $\epsilon$.
  • Figure 3: The distributions of density and temperature with $N=16$.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2