Efficient solutions of eigenvalue problems in rarefied gas flows

TL;DR

This paper introduces an efficient iterative method for solving eigenvalue problems in the linear stability analysis of the Boltzmann equation, enabling accurate and fast computations in rarefied gas flows relevant to space exploration.

Contribution

The paper presents a novel iterative approach combining shifted inverse power and synthetic schemes to efficiently solve high-dimensional eigenvalue problems in kinetic theory.

Findings

High efficiency and accuracy demonstrated in planar sound wave and Couette flow cases.

Eigenpairs computed with only a few hundred iterations.

Spatial cell size can be larger than the molecular mean free path in near-continuum regimes.

Abstract

The linear stability analysis of the Boltzmann kinetic equation has recently garnered research interest due to its potential applications in space exploration, where rarefaction effects can render the Navier Stokes equations invalid. Since the Boltzmann equation is defined in a seven-dimensional phase space, directly solving the associated eigenvalue problems is computationally intractable. In this paper, we propose an efficient iterative method to solve the linear stability equation of the kinetic equation. The solution process involves both outer and inner iterations. In the outer iteration, the shifted inverse power method is employed to compute selected eigenvalues and their corresponding eigenfunctions of interest. For the inner iteration, which involves inverting the high-dimensional system for the velocity distribution function, we adopt our recently developed general synthetic iterative scheme to ensure fast converging and asymptotic preserving properties. As a proof of concept, our method demonstrates both high efficiency and accuracy in planar sound wave and Couette flow. Each eigenpair can be computed with only a few hundred iterations of the kinetic equation, and the spatial cell size can be significantly larger than the molecular mean free path in near-continuum flow regimes.