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In this work, the Navier-Stokes (NS) solver is combined with the Directsimulation Monte Carlo (DSMC) solver in a direct way, under the wave-particleformulation [J. Comput. Phys. 401, 108977 (2020)]. Different from the classicaldomain decomposition method with buffer zone for overlap, in the proposeddirect unified wave-particle (DUWP) method, the NS solver is coupled with DSMCsolver on the level of algorithm. Automatically, in the rarefied flow regime,the DSMC solver leads the simulation, while the NS solver leads the continuumflow simulation. Thus advantages of accuracy and efficiency are both taken. Atinternal flow regimes, like the transition flow regime, the method is accurateas well because a kind of mesoscopic modeling is proposed in this work, whichgives the DUWP method the multi-scale property. Specifically, as to thecollision process, at $t < \tau$, it is supposed that only single collisionhappens, and the collision term of DSMC is just used. At $t > \tau$, it isderived that $1-\tau/\Delta t$ of particles should experience multiplecollisions, which will be absorbed into the wave part and calculated by the NSsolver. Then the DSMC and NS solver can be coupled in a direct and simple way,bringing about multi-scale property. The governing equation is derived andnamed as multi-scale Boltzmann equation. Different from the originalwave-particle method, in the proposed DUWP method, the wave-particleformulation is no more restricted by the Boltzmann-BGK type model and theenormous research findings of DSMC and NS solvers can be utilized into muchmore complicated flows, like the thermochemical non-equilibrium flow. In thiswork, one-dimensional cases in monatomic argon gas are preliminarily tested,such as shock structures and Sod shock tubes.

A direct unified wave-particle method for simulating non-equilibrium flows