Crank–Nicolson Bhatnagar–Gross–Krook integrator for multiscale particle-based kinetic simulations

Solving the Bhatnagar–Gross–Krook (BGK) equation with a stochastic particle approach enables efficient and flexible simulations of flows in the transition regime, between continuum and free molecular flow. However, the usual first-order operator splitting between particle movement and relaxation imposes restrictions on the time step, causing the computational cost to increase with the gas density. The Crank–Nicolson stochastic particle BGK (CN-SPBGK) method is introduced here as an advanced particle-based kinetic solver designed for multi-scale gas flow simulations. This method integrates the BGK equation with second-order accuracy across all Knudsen number regimes without requiring additional parameters, while asymptotically preserving the Navier–Stokes flux in the continuum regime. Comparisons with preexisting particle BGK methods are conducted on several test cases, with CN-SPBGK demonstrating more consistent convergence and accuracy.