A direct relaxation process for particle methods in gas-kinetic theory

The multi-scale flow mechanism is crucial for the force and heat loaded on near-space vehicles, the control of spacecraft, and the propelling and cooling of microelectromechanical systems. Since the continuum and rarefied flows often coexist, the prediction of multi-scale flow is complicated. One efficient way is constructing numerical methods by adopting the multi-scale temporal integral solutions (or characteristic line solutions) for model equations in the gas-kinetic theory. The model equations can be classified into the Fokker-Planck type and Bhatnagar-Gross-Krook type (BGK-type). Since these numerical methods are strictly based on model equations, they are also restricted by the model equations. The difficulty in constructing a model equation that has complete asymptotic preserving property for gas mixture with non-equilibrium internal energy will prevent the further extension of these methods. Therefore, this paper addresses the question whether a multi-scale numerical method can be established by directly adopting the relaxation rates of macroscopic variables, such as stress and heat flux, because these relaxation rates are the aggregate effect of particle collisions and are the essential constrains when constructing model equations. Since the particle-BGK method is concise, its collision term is replaced by the direct relaxation process, where the macroscopic variables first evolve according to their relaxation rates, and then, the after-collision molecules get their velocities from the after-evolution macroscopic variables. Therefore, the modified particle-BGK method does not depend on model equations. Finally, the validity and accuracy of the present method are examined with homogenous relaxation case, shock tube, shock structure, cavity flow, and hypersonic cylinder flow in transitional regime.