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

TL;DR

This paper presents a novel numerical solver for the steady-state Boltzmann equation that combines symmetric Gauss-Seidel, a preconditioned fixed-point iteration, and multigrid acceleration to improve convergence efficiency.

Contribution

The paper introduces a new solver integrating SGS, a preconditioned fixed-point iteration, and multigrid methods specifically for the steady-state Boltzmann equation.

Findings

Significant convergence speed-up with the preconditioned fixed-point iteration.

Effective acceleration of the solver using multigrid methods.

Validated through numerical experiments demonstrating improved performance.

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.