  A deterministic-stochastic coupling scheme is developed for simulating
rarefied gas flows, where the key process is the alternative solving of the
macroscopic synthetic equations [Su et al., J. Comput. Phys., 407 (2020)
109245] and the mesoscopic equation via the asymptotic-preserving time-relaxed
Monte Carlo scheme [Fei, J. Comput. Phys., 486 (2023) 112128]. Firstly, the
macroscopic synthetic equations are exactly derived from the Boltzmann
equation, incorporating not only the Newtonian viscosity and Fourier thermal
conduction laws but also higher-order constitutive relations that capture
rarefaction effects; the latter are extracted from the stochastic solver over a
defined sampling interval. Secondly, the macroscopic synthetic equations, with
the initial field extracted from the stochastic solver over the same sampling
interval, are solved to the steady state or over certain iteration steps.
Finally, the simulation particles in the stochastic solver are updated to match
the density, velocity, and temperature obtained from the macroscopic synthetic
equations. Moreover, simulation particles in the subsequent interval will be
partly sampled according to the solutions of macroscopic synthetic equations.
As a result, our coupling strategy enhances the asymptotic-preserving
characteristic of the stochastic solver and substantially accelerates
convergence towards the steady state. Several numerical tests are performed,
and it is found that our method can reduce the computational cost in the
near-continuum flow regime by two orders of magnitude compared to the direct
simulation Monte Carlo method.
