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Asymptotic Analysis of IMEX-RK Methods for ES-BGK Model at Navier-Stokes level

Sebastiano Boscarino, Seung Yeon Cho

TL;DR

The work addresses solving stiff kinetic equations by IMEX-RK time discretizations and aims to preserve the Navier–Stokes limit without resolving the small Knudsen scale $\varepsilon$. It provides a unified asymptotic analysis for IMEX-RK Type I and II schemes, deriving conditions under which they are asymptotically preserving and, with suitable coefficients, asymptotically accurate. A key contribution is the identification and demonstration of extra order conditions for Type II schemes that ensure uniform accuracy across the whole range of $\varepsilon$, exemplified by the IMEX-II-ISA3 scheme which mitigates order reduction in intermediate regimes. Numerical tests on BGK and ES-BGK models confirm that ISA3 achieves robust, high-order accuracy across $\varepsilon$, validating the theoretical findings and offering practical guidance for choosing schemes in kinetic-fluid coupling problems.

Abstract

Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further suggest some IMEX-RK methods developed in the literature to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.

Asymptotic Analysis of IMEX-RK Methods for ES-BGK Model at Navier-Stokes level

TL;DR

The work addresses solving stiff kinetic equations by IMEX-RK time discretizations and aims to preserve the Navier–Stokes limit without resolving the small Knudsen scale . It provides a unified asymptotic analysis for IMEX-RK Type I and II schemes, deriving conditions under which they are asymptotically preserving and, with suitable coefficients, asymptotically accurate. A key contribution is the identification and demonstration of extra order conditions for Type II schemes that ensure uniform accuracy across the whole range of , exemplified by the IMEX-II-ISA3 scheme which mitigates order reduction in intermediate regimes. Numerical tests on BGK and ES-BGK models confirm that ISA3 achieves robust, high-order accuracy across , validating the theoretical findings and offering practical guidance for choosing schemes in kinetic-fluid coupling problems.

Abstract

Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further suggest some IMEX-RK methods developed in the literature to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.

Paper Structure

This paper contains 11 sections, 10 theorems, 148 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. IMEX-RK schemes for ES-BGK equation
  3. Asymptotic Properties of the IMEX RK Schemes
  4. Preserving the Euler Limit.
  5. Preserving the Navier-Stokes Limit.
  6. Numerical Results
  7. Accuracy test.
  8. Conclusions
  9. Appendix
  10. Basic properties of Gaussian distribution function
  11. Chapmann-Enskog expansion of ES-BGK model

Key Result

Proposition 3.1

Consider the IMEX-RK method (IMEXcomp)-(solnum) of type I. Then in the limit $\varepsilon \to 0$, for a fixed $\Delta t$, the scheme becomes the explicit RK scheme characterized by the pair $(\tilde{A},\tilde{b})$ applied to the limit Euler equations (Eq:Euler).

Figures (4)

  • Figure 4.1: Comparison between reference solutions of BTE and NSEs with the numerical solutions for ES-BGK model. We report the case of density, velocity, temperature and heat flux at time $t=0.1,\,0.25,\,0.4$. The Knudsern number is $\varepsilon=0.5$.
  • Figure 4.2: Comparison between reference solutions of BTE and NSEs with the numerical solutions for ES-BGK model. We report the case of density, velocity, temperature and heat flux at time $t=0.1,\,0.25,\,0.4$. The Knudsern number is $\varepsilon=0.1$.
  • Figure 4.3: Comparison between reference solutions of Navier-Stokes equations and the numerical solutions of IMEX-II-GSA3 (left) and a zoom (right). Uniform $40 \times 40$ points for the velocity domain $[-20, 20] \times [-20, 20]$
  • Figure 4.4: The shear stress and heat flux for $\varepsilon = 10^{-2}, 10^{-4}$. Uniform $40 \times 40$ points for the velocity domain $[-20, 20] \times [-20, 20].$

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • Corollary 3.2
  • Remark 3.3
  • Proposition 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • ...and 10 more