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A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo, Seok-Bae Yun

TL;DR

The paper tackles efficient, accurate simulation of rarefied gas dynamics by solving the ellipsoidal BGK (ES-BGK) model with high-order conservative semi-Lagrangian schemes. It develops first-order and high-order SL discretizations using $L$-stable DIRK or BDF time stepping, along with conservative CWENO-based reconstruction and a weighted $L^2$-minimization to preserve mass, momentum, and energy, and it explicitly updates the temperature tensor to avoid Newton solvers. The authors prove conservation properties and analyze the asymptotic limit as the Knudsen number vanishes, showing convergence to a local Maxwellian and consistency with the Navier–Stokes equations under suitable assumptions. Numerical tests on accuracy, Riemann problems, and Lax shock tubes demonstrate high-order accuracy, correct kinetic-fluid transitions, and good agreement with NSE in the small-knudsen regime, validating both the method and its capacity to capture multi-regime gas dynamics. Overall, the work provides a robust computational framework for ES-BGK that preserves fundamental invariants, handles stiff relaxation efficiently, and bridges kinetic and hydrodynamic descriptions in a unified semi-Lagrangian setting.

Abstract

In this paper, we propose a high order conservative semi-Lagrangian scheme (SL) for the ellipsoidal BGK model of the Boltzmann transport equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator with small Knudsen number, we employ high order $L$-stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit schemes are designed to update solutions explicitly without resorting to any Newton solver. We present several numerical tests to demonstrate the accuracy and efficiency of the proposed methods. These methods allow us to obtain accurate approximations of the solutions to the Navier-Stokes equations or the Boltzmann equation for moderate or relatively large Knudsen numbers, respectively.

A conservative semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation

TL;DR

The paper tackles efficient, accurate simulation of rarefied gas dynamics by solving the ellipsoidal BGK (ES-BGK) model with high-order conservative semi-Lagrangian schemes. It develops first-order and high-order SL discretizations using -stable DIRK or BDF time stepping, along with conservative CWENO-based reconstruction and a weighted -minimization to preserve mass, momentum, and energy, and it explicitly updates the temperature tensor to avoid Newton solvers. The authors prove conservation properties and analyze the asymptotic limit as the Knudsen number vanishes, showing convergence to a local Maxwellian and consistency with the Navier–Stokes equations under suitable assumptions. Numerical tests on accuracy, Riemann problems, and Lax shock tubes demonstrate high-order accuracy, correct kinetic-fluid transitions, and good agreement with NSE in the small-knudsen regime, validating both the method and its capacity to capture multi-regime gas dynamics. Overall, the work provides a robust computational framework for ES-BGK that preserves fundamental invariants, handles stiff relaxation efficiently, and bridges kinetic and hydrodynamic descriptions in a unified semi-Lagrangian setting.

Abstract

In this paper, we propose a high order conservative semi-Lagrangian scheme (SL) for the ellipsoidal BGK model of the Boltzmann transport equation. To avoid the time step restriction induced by the convection term, we adopt the semi-Lagrangian approach. For treating the nonlinear stiff relaxation operator with small Knudsen number, we employ high order -stable diagonally implicit Runge-Kutta time discretization or backward difference formula. The proposed implicit schemes are designed to update solutions explicitly without resorting to any Newton solver. We present several numerical tests to demonstrate the accuracy and efficiency of the proposed methods. These methods allow us to obtain accurate approximations of the solutions to the Navier-Stokes equations or the Boltzmann equation for moderate or relatively large Knudsen numbers, respectively.

Paper Structure

This paper contains 14 sections, 5 theorems, 114 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Derivation of the SL method
  3. Notation
  4. Derivation of the first order scheme
  5. Derivation of the high order scheme
  6. DIRK methods
  7. BDF methods
  8. Weighted $L^2$-minimization for moment correction
  9. properties of the numerical method
  10. Numerical tests
  11. Accuracy test
  12. Riemann problem
  13. Lax shock tube problem
  14. Conclusion

Key Result

Proposition 3.1

Given $\varepsilon>0$ and $t^n > 0$, the first order methods satisfies

Figures (3)

  • Figure 1: A Riemann problem. Comparison of three solutions: BTE(blue) ES-BGK(red) NSE(black) for $\varepsilon=0.5$, CFL=$2$, $N_x=200$, $N_v=160$.
  • Figure 2: A Riemann problem. Comparison of three solutions: BTE(blue), ES-BGK(red), NSE(black) for $\varepsilon=0.1$, CFL=$2$, $N_x=200$, $N_v=96$.
  • Figure 3: Lax shock tube problem.

Theorems & Definitions (11)

  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 1 more