A Coupled IMEX Domain Decomposition Method for High-Order Time Integration of the ES-BGK Model of the Boltzmann Equation

TL;DR

This work develops a high-order IMEX domain-decomposition scheme for the ES-BGK Boltzmann model, dynamically separating equilibrium (Euler) and non-equilibrium (kinetic ES-BGK) regions. By coupling macroscopic and kinetic solvers at every Runge-Kutta stage and using Chapman-Enskog–based indicators, it preserves third-order temporal accuracy while achieving significant speedups. The approach demonstrates robust accuracy across 1D and 2D tests, correctly capturing shocks, instabilities, and boundary-layer phenomena, with kinetic regions confined to non-equilibrium zones. The framework is extensible to other kinetic problems and time integrators, offering a scalable, asymptotic-preserving method for multiscale gas dynamics.

Abstract

In this paper, we propose a high-order domain decomposition method for the ES-BGK model of the Boltzmann equation, which dynamically detects regions of equilibrium and non-equilibrium. Our implementation automatically switches between Euler equations in regions where the fluid is at equilibrium, and the ES-BGK model elsewhere. The main challenge addressed in this work is the development of a coupled strategy between the macroscopic and the kinetic solvers, which preserves the overall temporal order of accuracy of the scheme. A coupled IMEX method is introduced across decomposed subdomains and solvers. This approach is based on a coupled IMEX method and allows high accuracy and computational efficiency. Several numerical simulations in two space dimensions are performed, in order to validate the robustness of our approach and the expected temporal high-order convergence.

Figures (10)

• Figure 1: On the left, density at final time $t_{end}=0.86$ obtained with the smallest time step and Knudsen number $\varepsilon=10^{-6}$, using the hybrid scheme with high-order coupling. On the right, domain decomposition used in all simulations for testing the order.
• Figure 2: Convergence curves obtained using the full Euler Solver, the full Kinetic Solver and a comparison between the high-order coupling and low-order coupling, for three different values of the Knudsen number $\varepsilon=10^{-2}$ (on the left), $\varepsilon=10^{-4}$ (in the middle) and $\varepsilon=10^{-6}$ (on the right).
• Figure 3: Density, Temperature and Velocity evolution at $t=0.05$, $t=0.10$ and $t=0.15$ for the Sod shock tube with initial condition \ref{['eq::initial_Sod']}.
• Figure 4: In the first row domain adaptation is shown, whereas the other rows present the densities obtained using the hybrid (second row), the full Kinetic (third row) and full Euler (fourth row) solvers, respectively, for the Kelvin-Helmholtz instability with initial condition \ref{['eq::initial_Kelvin']}. The solutions are displayed at time $t=0.9$ and $t=1.7$.
• Figure 5: In the first row domain adaptation is shown, whereas the other rows present the densities obtained using the hybrid (second row), the full Kinetic (third row) and full Euler (fourth row) solvers, respectively, for the shock bubble interaction with initial condition \ref{['eq::initial_Bubble']}. The solutions are displayed at time $t=0.60$, $t=0.80$ and $t=1.5$.

Theorems & Definitions (3)

• Remark 3.1
• Remark 3.2
• Remark 3.3