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arXiv · 2509.21832

Micro-macro kinetic flux-vector splitting schemes for the multidimensional Boltzmann-ES-BGK equation

James A. Rossmanith·Preeti Sar
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Abstract

The kinetic Boltzmann equation models gas dynamics over a wide range of spatial and temporal scales. Simplified versions of the full Boltzmann collision operator, such as the classical Bhatnagar-Gross-Krook (BGK) and the closely related Ellipsoidal-Statistical-BGK (ES-BGK) operators, can dramatically reduce the computational cost of solving kinetic equations numerically. Classical BGK yields incorrect transport coefficients (relative to the full Boltzmann collision operator) at low Knudsen numbers, whereas ES-BGK captures them correctly. In this work, we develop a finite-volume method based on a micro-macro decomposition of the distribution function, which requires a smaller velocity mesh than direct kinetic methods for low and intermediate Knudsen numbers. The macro portion of the model is a fluid model with a moment closure derived from the heat-flux tensor calculated from the micro portion. The micro portion is obtained by applying to the original kinetic equation a projector into the orthogonal complement of the null space of the collision operator -- this projector depends on the macro portion. In particular, we extend the technique of Bennoune, Lemou, and Mieussens [{\it Uniformly stable schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comput. Phys. (2008)}] to two-space dimensions, the ES-BGK collision operator, and problems with reflecting wall boundary conditions. The collision operator in the micro and macro equations is handled via L-stable implicit time discretizations, while the transport terms are computed via kinetic flux vector splitting (for the macro equations) and upwind differencing (for the micro equation). The resulting scheme is applied to various test cases in 1D and 2D. The 2D version of the code is parallelized using MPI, and we present weak- and strong-scaling studies with varying numbers of processors.

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BibTeXRIS

James A. Rossmanith, Preeti Sar. 2026-07-15. Micro-macro kinetic flux-vector splitting schemes for the multidimensional Boltzmann-ES-BGK equation. https://arxiv.org/abs/2509.21832

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