A wave-particle decomposition framework for multiscale kinetic transport: continuous-spectrum equations and coupled wave-particle iteration
Abstract
The wave-particle decomposition is presented as a framework for multiscale kinetic transport, comprising a theory and an algorithm. Along the characteristics of the kinetic equation, the equilibrium integral of the solution over a local kinetic horizon defines a wave, which depends only on the conservative variables, and its exact complement defines a particle. Both obey kinetic equations whose moments are extended Navier-Stokes equations, and the conservation laws together with the particle equation form a closed system that is equivalent to the kinetic equation for every horizon. Parametrized by the ratio of the horizon to the relaxation time, these systems form a continuous spectrum from kinetic theory to Navier-Stokes hydrodynamics that adapts to the flow from cell to cell. The algorithm is a coupled wave-particle iteration for steady flows. It alternates a macroscopic iteration for the conservative variables, with the particle frozen, and a kinetic iteration for the particle, with the wave fixed. Its fixed point is the discrete kinetic solution, it is asymptotic preserving, and its near-continuum convergence factor is bounded by the share of the transport carried by the particle, a share that decays exponentially with the horizon-to-relaxation ratio. The collision model enters only through a conservative remainder, and the equations and the iteration are written out for the Bhatnagar-Gross-Krook, Shakhov and ellipsoidal statistical models, the Boltzmann and Landau equations, neutron transport and radiative transfer. The decomposition is related to the unified gas-kinetic wave-particle method, the micro-macro decomposition, penalization and synthetic acceleration. Computations of hypersonic flows past a cylinder and around the three-dimensional X38 vehicle illustrate how the division of the distribution adapts to the flow and how strongly the iteration is accelerated.
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BibTeX
@misc{liu2026waveparticle,
title = {{A wave-particle decomposition framework for multiscale kinetic transport: continuous-spectrum equations and coupled wave-particle iteration}},
author = {Chang Liu and Kun Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33622},
eprint = {2609.33622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33622},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A wave-particle decomposition framework for multiscale kinetic transport: continuous-spectrum equations and coupled wave-particle iteration
AU - Chang Liu
AU - Kun Xu
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33622
UR - https://arxiv.org/abs/2609.33622
AB - The wave-particle decomposition is presented as a framework for multiscale kinetic transport, comprising a theory and an algorithm. Along the characteristics of the kinetic equation, the equilibrium integral of the solution over a local kinetic horizon defines a wave, which depends only on the conservative variables, and its exact complement defines a particle. Both obey kinetic equations whose moments are extended Navier-Stokes equations, and the conservation laws together with the particle equation form a closed system that is equivalent to the kinetic equation for every horizon. Parametrized by the ratio of the horizon to the relaxation time, these systems form a continuous spectrum from kinetic theory to Navier-Stokes hydrodynamics that adapts to the flow from cell to cell. The algorithm is a coupled wave-particle iteration for steady flows. It alternates a macroscopic iteration for the conservative variables, with the particle frozen, and a kinetic iteration for the particle, with the wave fixed. Its fixed point is the discrete kinetic solution, it is asymptotic preserving, and its near-continuum convergence factor is bounded by the share of the transport carried by the particle, a share that decays exponentially with the horizon-to-relaxation ratio. The collision model enters only through a conservative remainder, and the equations and the iteration are written out for the Bhatnagar-Gross-Krook, Shakhov and ellipsoidal statistical models, the Boltzmann and Landau equations, neutron transport and radiative transfer. The decomposition is related to the unified gas-kinetic wave-particle method, the micro-macro decomposition, penalization and synthetic acceleration. Computations of hypersonic flows past a cylinder and around the three-dimensional X38 vehicle illustrate how the division of the distribution adapts to the flow and how strongly the iteration is accelerated.
ER -