An IMEX multi-level domain-decomposition method for the Boltzmann equation on unstructured meshes

Abstract

We introduce a dynamically adaptive multi-level method for the multiscale Boltzmann equation on unstructured meshes, coupling the compressible Euler, Navier-Stokes, and full Boltzmann descriptions within a single asymptotic-preserving IMEX framework. Unlike standard domain-decomposition approaches, where independent solvers exchange data only through interface conditions, the three models form a unified multiregime scheme. They share the same globally stiffly accurate IMEX Runge-Kutta discretization and are synchronized at every internal stage through kinetic-to-fluid moment projections and fluid-to-kinetic lifting. This stage-wise coupling maintains temporal consistency across model interfaces, while the asymptotic-preserving structure ensures compatibility with the fluid limit as the Knudsen number becomes small. The active description is selected in each cell through moment-realizability indicators derived from successive Chapman-Enskog approximations. The hierarchy uses Euler in near-equilibrium regions, Navier-Stokes where first-order nonequilibrium corrections are sufficient, and the full Boltzmann equation where a kinetic description is required. Using Navier-Stokes as the intermediate level avoids evolving a velocity distribution in mildly nonequilibrium regions, which is particularly advantageous on unstructured meshes, where kinetic transport dominates the cost. The spatial discretization combines finite-volume WENO reconstruction on triangular meshes with a conservative fast spectral method for the Boltzmann collision operator. Numerical experiments show that the method closely reproduces the full Boltzmann reference while significantly reducing the computational cost. These results suggest that tightly coupled AP model hierarchies are a viable alternative to interface-based domain decomposition for deterministic kinetic simulations in complex geometries.