A temporal hybrid algorithm for coupling direct simulation Monte Carlo with a stochastic particle method

The direct simulation Monte Carlo (DSMC) method provides a particle-based numerical solution to the Boltzmann equation and is widely employed for simulating rarefied nonequilibrium gas flows. With advances in aerospace engineering and micro/nano-scale technologies, gas flows increasingly exhibit coexistence of rarefied and continuum/near-continuum regimes, which calls for larger time steps and coarser spatial grids to achieve efficient and accurate numerical simulations. However, the mesh sizes and time steps in DSMC are constrained by the single-scale nature of the Boltzmann equation and the explicit treatment of the collision term following operator splitting. Recently, a multi-scale Boltzmann equation was proposed, providing a more consistent coupling between the DSMC method and the Navier–Stokes equations. Inspired by this multi-scale equation, a novel multi-scale hybrid strategy and the corresponding particle method are proposed, which couple the stochastic particle (SP) Shakhov with DSMC based on physical scales evaluated by the observation time (Δtc, numerical time step) and the relaxation time (τ, the same order as the mean collision time). As a numerical representation of the mechanism, the mathematically weighted average of the Boltzmann and Shakhov collision terms is transformed into a collision process decomposed into sequential DSMC collisions and large-time SP-Shakhov collisions with the time step ΔtG split into τΔtcΔtG for DSMC and (1−τΔtc)ΔtG for SP. The proposed hybrid method combines the advantages of the DSMC and SP-Shakhov methods, maintaining the accuracy of DSMC in rarefied flows while preserving the efficiency of SP in near-continuum flows. The validity and accuracy of the proposed method are demonstrated through a series of benchmark cases, including one-dimensional Sod shock tube, two-dimensional hypersonic flow around a cylinder and jet expansion into the vacuum, and three-dimensional hypersonic flow around a sphere and an X-38-like vehicle in near-continuum flow regimes.