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Symmetric Gauss-Seidel Method with a Preconditioned Fixed-Point Iteration for the Steady-State Boltzmann equation

Zhenning Cai, Xiaoyu Dong, Jingwei Hu

TL;DR

This work addresses efficiently solving the steady-state Boltzmann equation for rarefied gases across regimes by developing a solver based on a symmetric Gauss-Seidel (SGS) framework augmented with a preconditioned fixed-point inner solve that leverages the asymptotic limit. A nonlinear multigrid approach further accelerates convergence, and the method extends to quadratic collision operators via a BGK penalty, using a discrete Maxwellian to preserve moments and a Fourier spectral method for collisions. Across BGK and Maxwell-molecule collisions, the proposed SGS-PFP method shows substantial reductions in outer iterations and wall-clock time, especially for small Knudsen numbers ($ ext{ε}$), with multigrid providing strong acceleration in 2D/3D settings. The approach delivers a practical, scalable kinetic solver that avoids macroscopic preconditioning while achieving robust convergence across regimes, making it attractive for high-fidelity rarefied-gas simulations.

Abstract

We introduce a numerical solver for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. To solve the nonlinear system on each grid cell derived from the SGS method, a fixed-point iteration preconditioned with its asymptotic limit is developed. The preconditioner only requires solving an algebraic system which is easy to implement and can speed up the convergence significantly especially in the case of small Knudsen numbers. Additionally, we couple our numerical scheme with the multigrid method to accelerate convergence. A variety of numerical experiments are carried out to illustrate the effectiveness of these methods.

Symmetric Gauss-Seidel Method with a Preconditioned Fixed-Point Iteration for the Steady-State Boltzmann equation

TL;DR

This work addresses efficiently solving the steady-state Boltzmann equation for rarefied gases across regimes by developing a solver based on a symmetric Gauss-Seidel (SGS) framework augmented with a preconditioned fixed-point inner solve that leverages the asymptotic limit. A nonlinear multigrid approach further accelerates convergence, and the method extends to quadratic collision operators via a BGK penalty, using a discrete Maxwellian to preserve moments and a Fourier spectral method for collisions. Across BGK and Maxwell-molecule collisions, the proposed SGS-PFP method shows substantial reductions in outer iterations and wall-clock time, especially for small Knudsen numbers (), with multigrid providing strong acceleration in 2D/3D settings. The approach delivers a practical, scalable kinetic solver that avoids macroscopic preconditioning while achieving robust convergence across regimes, making it attractive for high-fidelity rarefied-gas simulations.

Abstract

We introduce a numerical solver for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. To solve the nonlinear system on each grid cell derived from the SGS method, a fixed-point iteration preconditioned with its asymptotic limit is developed. The preconditioner only requires solving an algebraic system which is easy to implement and can speed up the convergence significantly especially in the case of small Knudsen numbers. Additionally, we couple our numerical scheme with the multigrid method to accelerate convergence. A variety of numerical experiments are carried out to illustrate the effectiveness of these methods.
Paper Structure (16 sections, 47 equations, 19 figures, 5 tables)

This paper contains 16 sections, 47 equations, 19 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Discretization of the steady-state Boltzmann equation
  3. Spatial discretization
  4. Velocity discretization
  5. Iterative method for the steady-state Boltzmann equation
  6. Symmetric Gauss-Seidel method for the BGK equation
  7. A preconditioned fixed-point iteration for the nonlinear equation
  8. Numerical method for the Boltzmann equation with quadratic collision operators
  9. Numerical experiments
  10. BGK collision model
  11. One-dimensional heat transfer
  12. Heat transfer in a cavity
  13. Binary collision operator
  14. Heat transfer between parallel plates
  15. Lid-driven cavity flow
  16. ...and 1 more sections

Figures (19)

  • Figure 1: The distributions of density and temperature for one-dimensional heat transfer.
  • Figure 2: The errors of upwind finite volume schemes with various $\epsilon$ for one-dimensional heat transfer.
  • Figure 3: The convergence of iterative methods for one-dimensional heat transfer.
  • Figure 4: The average inner iterations of iterative methods (1st-order) for one-dimensional heat transfer.
  • Figure 5: The average inner iterations of iterative methods (2nd-order) for one-dimensional heat transfer.
  • ...and 14 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4