Boosting the convergence of DSMC by GSIS

Abstract: 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.