Hierarchical dynamic domain decomposition for the multiscale Boltzmann equation

TL;DR

The article develops a hierarchical, three-regime solver that adaptively couples Euler, ES-BGK, and full Boltzmann dynamics to efficiently simulate multiscale kinetic flows in 2D space with 3D velocity. Central to the approach are the moment realizability criterion and the Chapman-Enskog expansion, which drive regime transitions, and asymptotic-preserving time integrators paired with fast Fourier-based Boltzmann solvers. The method achieves high accuracy in shocks and non-equilibrium regions while dramatically reducing cost compared to full Boltzmann simulations, with demonstrated speedups of roughly 1.3–3× across benchmark tests. This framework enables robust, scalable simulations on complex geometries and paves the way for extensions to multi-species and HPC-accelerated implementations.

Abstract

In this work, we present a hierarchical domain decomposition method for the multi-scale Boltzmann equation based on moment realizability matrices, a concept introduced by Levermore, Morokoff, and Nadiga in \cite{lev-mor-nad-1998}. This criterion is used to dynamically partition the two-dimensional spatial domain into three regimes: the Euler regime, an intermediate kinetic regime governed by the ES-BGK model, and the full Boltzmann regime. The key advantage of this approach lies in the use of Euler equations in regions where the flow is near hydrodynamic equilibrium, the ES-BGK model in moderately non-equilibrium regions where a fluid description is insufficient but full kinetic resolution is not yet necessary, and the full Boltzmann solver where strong non-equilibrium effects dominate, such as near shocks and boundary layers. This allows for both high accuracy and significant computational savings, as the Euler solver and the ES-BGK models are considerably cheaper than the full kinetic Boltzmann model. To ensure accurate and efficient coupling between regimes, we employ asymptotic-preserving (AP) numerical schemes and fast spectral solvers for evaluating the Boltzmann collision operator. Among the main novelties of this work are the use of a full 2D spatial and 3D velocity decomposition, the integration of three distinct physical regimes within a unified solver framework, and a parallelized implementation exploiting CPU multithreading. This combination enables robust and scalable simulation of multiscale kinetic flows with complex geometries.

Figures (9)

• Figure 1: Example of schematic view of the stencil for the computation of the first and second spatial derivatives, and for the application of a finite volume method.
• Figure 2: Time evolution of density for \ref{['subTestSod']}. In the first row the regime adaptation and the density at three different times are displayed. Red, green and black dots, identify, respectively, cells updated using Euler equations, Boltzmann equations with ES-BGK operator and Boltzmann equations with Boltzmann (Hard-Sphere) operator, using the new hybrid scheme. The cyan line is the analytical solution, the magenta one is the solution obtained using only the Euler equations, and the blue line is computed using only the Boltzmann equation. The same plots, but for the temperature, are showed in the second row. The time evolution of the density for the hybrid scheme in the 2D space (solution constant along the $y-$direction) is displayed in the last row.
• Figure 3: Time evolution of density for \ref{['subTestFluidFixedRectObs']}. The first row displays the regime adaptation at three different times. Red, green and black dots, identify, respectively, cells updated using Euler equations, ES-BGK equation and Boltzmann equation. In the second, third and fourth row are displayed, respectively, the time evolution of density computed using hybrid scheme, Euler equations and Boltzmann equation.
• Figure 4: Time evolution of temperature for \ref{['subTestFluidFixedRectObs']}. In the first, second and third row are displayed, respectively, the time evolution of temperature computed using hybrid scheme, Euler equations and Boltzmann equation.
• Figure 5: Time evolution of density for \ref{['subTestFluidMovingRest']}. The first row displays the regime adaptation at three different times. Red, green and black dots, identify, respectively, cells updated using Euler equations, ES-BGK equation and Boltzmann equation. In the second, third and fourth row are displayed, respectively, the time evolution of density computed using hybrid scheme, Euler equations and Boltzmann equation.

Theorems & Definitions (3)

• Remark 1
• Proposition 4.1
• Lemma 4.1