Implementing the direct relaxation process in the stochastic particle method for flexible molecular collisions.

This article focuses on developing and validating a stochastic particle method with a direct relaxation (DR) process, denoted as SP-DR, to overcome limitations of traditional Boltzmann model equations in simulating multi-scale gas flows. The DR process directly uses relaxation rates of non-equilibrium macroscopic variables, such as stress tensor and heat flux, to model particle collisions without constructing explicit collision operators, providing flexibility to recover various kinetic models including Xu's and Yuan's velocity-dependent relaxation time models. The SP-DR method incorporates a generalized inlet/outlet boundary condition capable of handling subsonic, supersonic, and hypersonic flows simultaneously. Validation through multiple test cases—such as normal shock wave structures, Sod shock tube, cavity flow, supersonic/hypersonic flow past cylinders, and micro-channel flow—demonstrates that SP-DR accurately captures multi-scale flow physics and improves upon traditional BGK-type models, particularly in resolving temperature profile deviations in shock structures. The study concludes that the DR strategy offers a universal and efficient framework for particle-based kinetic simulations across different flow regimes and model equations.