A relaxation system for the low-Mach limit of kinetic equations: Stability and higher order asymptotic preserving scheme
Abstract
This work introduces a hyperbolic relaxation system designed for the simulation of kinetic equations in the low-Mach number limit. The methodology is built upon a micro-macro decomposition of the scaled BGK model, which reformulates the kinetic distribution function into a coupled system consisting of a macroscopic equilibrium part and a microscopic non-equilibrium remainder. By projecting the microscopic deviations onto a set of orthogonal polynomials, we derive a closed moment relaxation system. The resulting system is a version of Grad's 13 moment system with a linear hyperbolic part and a relaxation adapted to the incompressible limit. We prove the model's structural stability in the incompressible Navier-Stokes limit. Moreover, we develop a high order Asymptotic-Preserving (AP) numerical framework using Implicit-Explicit (IMEX) Runge Kutta schemes for temporal accuracy and finite difference WENO reconstructions as well as central difference approximations for high order spatial resolution. The proposed scheme ensures uniform stability and consistency across different physical regimes, automatically degenerating into a consistent high order discretization of the incompressible thermal Navier-Stokes limit as the scaling parameter vanishes. Numerical experiments in one and two dimensions corroborate the theoretical findings, demonstrating significant reductions in computational cost and robust performance across a wide range of Knudsen and Mach numbers.
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BibTeX
@misc{dimarco2026relaxation,
title = {{A relaxation system for the low-Mach limit of kinetic equations: Stability and higher order asymptotic preserving scheme}},
author = {Giacomo Dimarco and Axel Klar and Theresa Köfler and Lorenzo Pareschi and Sudarshan Tiwari and Yizhou Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00811},
eprint = {2610.00811},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00811},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A relaxation system for the low-Mach limit of kinetic equations: Stability and higher order asymptotic preserving scheme
AU - Giacomo Dimarco
AU - Axel Klar
AU - Theresa Köfler
AU - Lorenzo Pareschi
AU - Sudarshan Tiwari
AU - Yizhou Zhou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00811
UR - https://arxiv.org/abs/2610.00811
AB - This work introduces a hyperbolic relaxation system designed for the simulation of kinetic equations in the low-Mach number limit. The methodology is built upon a micro-macro decomposition of the scaled BGK model, which reformulates the kinetic distribution function into a coupled system consisting of a macroscopic equilibrium part and a microscopic non-equilibrium remainder. By projecting the microscopic deviations onto a set of orthogonal polynomials, we derive a closed moment relaxation system. The resulting system is a version of Grad's 13 moment system with a linear hyperbolic part and a relaxation adapted to the incompressible limit. We prove the model's structural stability in the incompressible Navier-Stokes limit. Moreover, we develop a high order Asymptotic-Preserving (AP) numerical framework using Implicit-Explicit (IMEX) Runge Kutta schemes for temporal accuracy and finite difference WENO reconstructions as well as central difference approximations for high order spatial resolution. The proposed scheme ensures uniform stability and consistency across different physical regimes, automatically degenerating into a consistent high order discretization of the incompressible thermal Navier-Stokes limit as the scaling parameter vanishes. Numerical experiments in one and two dimensions corroborate the theoretical findings, demonstrating significant reductions in computational cost and robust performance across a wide range of Knudsen and Mach numbers.
ER -