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Perfectly matched layers for the Boltzmann equation: stability and sensitivity analysis

Marco Sutti, Jan S. Hesthaven

TL;DR

The paper addresses truncation of unbounded kinetic problems by integrating a BGK-based absorbing layer through a perfectly matched layer (PML). It combines stability analysis—via energy decay and continued fractions on the symbol of the BGK+PML operator—with ANOVA-based sensitivity analysis to rank parameter importance, applying the methods to two test cases including an isentropic vortex. Key findings show that setting $ ext{λ}_0 = ext{λ}_1 = 0$ is necessary for stability, while the damping parameters (the PML thickness $L$ and exponent $eta$) dominate the absorbing layer's effectiveness, with minimal sensitivity to other parameters or functional choice. The work provides practical guidance for robust BGK+PML implementations and lays groundwork for coupling BGK-based PML regions with Navier–Stokes in future extensions. Altogether, the study advances reliable absorbing-layer design for kinetic models and demonstrates a systematic, data-driven way to identify critical PML parameters.

Abstract

We study the stability and sensitivity of an absorbing layer for the Boltzmann equation by examining the Bhatnagar-Gross-Krook (BGK) approximation and using the perfectly matched layer (PML) technique. To ensure stability, we discard some parameters in the model and calculate the total sensitivity indices of the remaining parameters using the ANOVA expansion of multivariate functions. We conduct extensive numerical experiments on two test cases to study stability and compute the total sensitivity indices, which allow us to identify the essential parameters of the model.

Perfectly matched layers for the Boltzmann equation: stability and sensitivity analysis

TL;DR

The paper addresses truncation of unbounded kinetic problems by integrating a BGK-based absorbing layer through a perfectly matched layer (PML). It combines stability analysis—via energy decay and continued fractions on the symbol of the BGK+PML operator—with ANOVA-based sensitivity analysis to rank parameter importance, applying the methods to two test cases including an isentropic vortex. Key findings show that setting is necessary for stability, while the damping parameters (the PML thickness and exponent ) dominate the absorbing layer's effectiveness, with minimal sensitivity to other parameters or functional choice. The work provides practical guidance for robust BGK+PML implementations and lays groundwork for coupling BGK-based PML regions with Navier–Stokes in future extensions. Altogether, the study advances reliable absorbing-layer design for kinetic models and demonstrates a systematic, data-driven way to identify critical PML parameters.

Abstract

We study the stability and sensitivity of an absorbing layer for the Boltzmann equation by examining the Bhatnagar-Gross-Krook (BGK) approximation and using the perfectly matched layer (PML) technique. To ensure stability, we discard some parameters in the model and calculate the total sensitivity indices of the remaining parameters using the ANOVA expansion of multivariate functions. We conduct extensive numerical experiments on two test cases to study stability and compute the total sensitivity indices, which allow us to identify the essential parameters of the model.
Paper Structure (36 sections, 4 theorems, 99 equations, 3 figures, 12 tables)

This paper contains 36 sections, 4 theorems, 99 equations, 3 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline of the paper
  4. Review of the BGK model
  5. Approximation of the BGK model
  6. Hyperbolicity
  7. Relationship with the Navier--Stokes equations
  8. A PML for the BGK model
  9. Damping functions
  10. Perfect matching
  11. Our study case: TEXT
  12. Implementation aspects and testing
  13. Spatial discretization
  14. Time discretization
  15. The CFL condition
  16. ...and 21 more sections

Key Result

Theorem 5.1

A necessary condition for well-posedness of eq:PDE_system_short is that, for all $k$, the eigenvalues $\lambda$ of $\hat{P}(\mathrm{i} k)$ satisfy the inequality $\mathrm{Re}(\lambda) \leqslant \alpha$, with $\alpha$ being a positive constant.

Figures (3)

  • Figure 1: Contour plots of the density distribution at different simulation times.
  • Figure 2: Time evolution of err-$a_{1}$. Panel (a): For several values of $\lambda_{0}$. Panel (b): For several PML thicknesses $L$.
  • Figure 3: BGK+PML model for the isentropic vortex test case. Contours of the $v$-velocity distribution from $-0.25$ to $0.25$ with an increment of $0.0125$ excluding the zero level, at four different time instants.

Theorems & Definitions (8)

  • Definition 2.1: evans2010partial
  • Definition 2.2: evans2010partial
  • Theorem 5.1: The Petrovskii condition Gustafsson1995
  • Theorem 5.2: Gustafsson1995
  • Theorem 5.3: Frank, 1946 Marden1966
  • Definition 6.1
  • Definition 6.2
  • Theorem 6.1: Wang:2003