Table 1 Outline of the efficient algorithm of the USP-BGK method

1. Initialization

Introduce initial computational particles in the computational domain. Their velocities are sampled from the initial auxiliary PDF \( \overset{\frown }{f}\left(\mathbf{c};\mathbf{x},0\right) \).

2. Streaming

Move the computational particles with their velocities and apply boundary conditions to obtain \( {\tilde{f}}^{\ast}\left(\mathbf{c};\mathbf{x},\Delta t\right) \).

3. Collision

(1 − e^−Δt/ε) part of particles are randomly selected from the cell to assign new velocities, which are sampled from the PDF f_U; the velocities of the remaining part of particles are unchanged. f_U is calculated based on Eqs. (12) and (19a, 19b). Their mean velocity and temperature are obtained based on particle tracking interpolation as shown in Eqs. (27) and (28), respectively. \( {\sigma}_{ij}^{\ast } \) and \( {q}_i^{\ast } \) use the average values of the computational cell, which are obtained according to Appendix.

After the collision step, the PDF of the computational particles is equal to \( \overset{\frown }{f}\left(\mathbf{c};\mathbf{x},\Delta t\right) \) and prepared for the next time step.

4. Sampling

Sample the macroscopic quantities (also see Appendix).