## Abstract

This paper investigates the convergence proof of the Direct Simulation Monte Carlo (DSMC) method and the Gas-Kinetic Unified Algorithm in simulating the Boltzmann equation. It can be shown that the particle velocity distribution function obtained by the DSMC method converges to a modified form of the Boltzmann equation, which is the equation of the gas-kinetic unified algorithm to directly solve the molecular velocity distribution function. Their convergence is derived through mathematical treatment. The collision frequency is presented using various molecular models and the local equilibrium distribution function is obtained by Enskog expansion using the converged equation of the DSMC method. These two expressions agree with those used in the unified algorithm. Numerical validation of the converging consistency between these two approaches is illustrated by simulating the pressure driven Poiseuille flow in the slip transition flow regime and the two-dimensional and three-dimensional flows around a circular cylinder and spherical-cone reentry body covering the whole flow regimes from low speed micro-channel flow to high speed non-equilibrium aerothermodynamics.

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Li, Z., Fang, M., Jiang, X. *et al.* Convergence proof of the DSMC method and the Gas-Kinetic Unified Algorithm for the Boltzmann equation.
*Sci. China Phys. Mech. Astron.* **56**, 404–417 (2013). https://doi.org/10.1007/s11433-013-4999-3

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DOI: https://doi.org/10.1007/s11433-013-4999-3