A high-order unified stochastic particle method based on the Bhatnagar-Gross-Krook model for multi-scale gas flows

Abstract

Stochastic particle methods, such as Direct Simulation Monte Carlo (DSMC), are successfully used to simulate rarefied gas flows. However, since traditional stochastic particle methods employ the operator splitting scheme, i.e., decoupling the molecular motion and collisions, they are of low order accuracy in the continuum regime. In contrast, the discrete velocity method can easily construct high order schemes, but its computational and memory cost is quite high due to the huge number of discrete velocity nodes, especially for the high Mach number flows. To improve the efficiency and accuracy of the simulation for multi-scale flows, a high order stochastic particle BGK method was developed in this paper. The DIRK (Diagonally Implicit Runge–Kutta) and WLS-ENO (weighted-least-squares based essentially non-oscillatory) schemes were used to achieve a high order of accuracy in time evolution and space interpolation, respectively. However, the implementation of DIRK and WLS-ENO with computational particles is not straightforward due to the statistical nature of the stochastic particle methods. To circumvent these difficulties, firstly, the micro-macro decomposition of the relaxation term of the BGK model, which was proposed in the unified stochastic particle BGK (USP-BGK) method, was employed. The fluid limit part of the relaxation term was treated following the DIRK scheme, while the kinetic part was solved with an exact integration scheme similar to the traditional stochastic particle BGK (SP-BGK) method. Therefore, in the continuum regime, the present stochastic particle method has the same asymptotic preserving and asymptotic accurate properties as the high order semi-Lagrangian method; and in the rarefied regime, it reduces to the traditional SP-BGK method. Additionally, to overcome the effect of thermal noise in the space interpolation, a new discontinuity indicator, which considers the influence of the thermal fluctuation, was also proposed for the WLS-ENO scheme. Several 1D and 2D benchmark problems with different Mach and Knudsen numbers were tested and analyzed, such as shock and sine wave interaction, Taylor-Green vortex flow and 2D Riemann problem. Compared to the traditional SP-BGK method, this proposed method can achieve a higher order of accuracy and significantly improve the computational efficiency in the continuum regime.