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Asymptotic-preserving semi-Lagrangian discontinuous Galerkin schemes for the Boltzmann equation

Xiaofeng Cai, Zhen Hao, Liu Liu, Jiayu Wan

TL;DR

The paper develops an asymptotic-preserving semi-Lagrangian discontinuous Galerkin (AP-SLDG) scheme for the Boltzmann equation to efficiently handle multi-scale transport without stringent time-step restrictions. A Shu-Osher form of IMEX-RK time stepping is used to define a robust moments update that recovers the Euler limit as $ε→0$, supported by order-condition analysis and a hypocoercivity-based stability framework. A BGK penalization is integrated to treat stiff collisions implicitly, and a novel moments-update procedure ensures positivity and asymptotic accuracy. Numerical experiments in 1D2V settings validate accuracy, AP behavior, and robustness across regimes, while highlighting order limitations for certain high-order IMEX-RK tableaux and the need for positivity-cognizant time-step constraints. The results advance efficient, high-order kinetic simulations across kinetic-to-fluid transitions and pave the way for adaptive, multi-scale schemes in complex geometries.

Abstract

In this work, we present an asymptotic-preserving semi-Lagrangian discontinuous Galerkin scheme for the Boltzmann equation that effectively handles multi-scale transport phenomena. The main challenge lies in designing appropriate moments update for penalization within the semi-Lagrangian framework. Inspired by [M. Ding, J. M. Qiu, and R. Shu, Multiscale Model. Simul. 21 (2023), no. 1, 143--167], the key ingredient is utilizing the Shu-Osher form of the scheme in the implicit-explicit Runge-Kutta (IMEX-RK) setting, which enables us to capture the correct limiting system by constructing an appropriate moments update procedure. Our theoretical analysis establishes accuracy order conditions for both the IMEX-RK time integration and the new moments update step. We also employ hypocoercivity techniques to establish stability for the linearized model. Numerical experiments for various test problems validate our proposed scheme's accuracy, asymptotic-preserving property, and robustness in various regimes, which demonstrates its effectiveness for multi-scale kinetic simulations.

Asymptotic-preserving semi-Lagrangian discontinuous Galerkin schemes for the Boltzmann equation

TL;DR

The paper develops an asymptotic-preserving semi-Lagrangian discontinuous Galerkin (AP-SLDG) scheme for the Boltzmann equation to efficiently handle multi-scale transport without stringent time-step restrictions. A Shu-Osher form of IMEX-RK time stepping is used to define a robust moments update that recovers the Euler limit as , supported by order-condition analysis and a hypocoercivity-based stability framework. A BGK penalization is integrated to treat stiff collisions implicitly, and a novel moments-update procedure ensures positivity and asymptotic accuracy. Numerical experiments in 1D2V settings validate accuracy, AP behavior, and robustness across regimes, while highlighting order limitations for certain high-order IMEX-RK tableaux and the need for positivity-cognizant time-step constraints. The results advance efficient, high-order kinetic simulations across kinetic-to-fluid transitions and pave the way for adaptive, multi-scale schemes in complex geometries.

Abstract

In this work, we present an asymptotic-preserving semi-Lagrangian discontinuous Galerkin scheme for the Boltzmann equation that effectively handles multi-scale transport phenomena. The main challenge lies in designing appropriate moments update for penalization within the semi-Lagrangian framework. Inspired by [M. Ding, J. M. Qiu, and R. Shu, Multiscale Model. Simul. 21 (2023), no. 1, 143--167], the key ingredient is utilizing the Shu-Osher form of the scheme in the implicit-explicit Runge-Kutta (IMEX-RK) setting, which enables us to capture the correct limiting system by constructing an appropriate moments update procedure. Our theoretical analysis establishes accuracy order conditions for both the IMEX-RK time integration and the new moments update step. We also employ hypocoercivity techniques to establish stability for the linearized model. Numerical experiments for various test problems validate our proposed scheme's accuracy, asymptotic-preserving property, and robustness in various regimes, which demonstrates its effectiveness for multi-scale kinetic simulations.
Paper Structure (16 sections, 10 theorems, 58 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 10 theorems, 58 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. The nodal AP-SLDG method for the Boltzmann equation
  4. Nodal DG framework
  5. Characteristic Galerkin weak formulation
  6. The AP-SLDG scheme
  7. High-order AP-SLDG schemes
  8. Moments update
  9. The final scheme
  10. Numerical analysis for the scheme
  11. Positivity-preserving property
  12. Asymptotic-preserving property
  13. Conditions for the accuracy order
  14. Stability analysis
  15. Numerical examples
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $I$ be a line segment on the interval $\Omega_x$, assume $f$ and $\psi$ are solutions of eq:BoltzmannEquationODE and eq:adjoint_problem respectively, then the following identity holds

Figures (5)

  • Figure 1: The time evolution of $||f - M_f||_{\ell^1}$ for ARS443 and DP2A242 in Test II.
  • Figure 2: Density, the first component of mean velocity and temperature of Test III, with CFL$=0.5$.
  • Figure 3: Density, the first component of mean velocity and temperature of Test III, with CFL$=2$.
  • Figure 4: Density, the first component of mean velocity and temperature of Test IV at $t=0.3$, with CFL = 0.5.
  • Figure 5: Density, the first component of mean velocity and temperature of Test IV at $t=0.3$, with CFL = 2.

Theorems & Definitions (21)

  • Proposition 3.1
  • Proof 1
  • Proposition 3.2
  • Remark 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Remark 3.6
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 11 more