Exponential Bhatnagar-Gross-Krook integrator for multiscale particle-based kinetic simulations

Despite the development of an extensive toolbox of multiscale rarefied flow simulators, such simulations remain challenging due to the significant disparity of collisional and macroscopic spatiotemporal scales. Our study offers a novel and consistent numerical scheme for a coupled treatment of particles advection and collision governed by the BGK evolution, honoring positivity of the velocity distribution. Our method shares its framework, in spirit, with the unified gas kinetic class of multiscale schemes. Yet it provides attractive features for particle-based stochastic simulations, readily implementable to existing direct simulation Monte Carlo codes. Two main innovations are integrated in the presented BGK particle method. The first ingredient is a high-order time integration that can be interpreted probabilistically, independent of the time step size. The next one is identifying modified particle distributions that remain invariant under the advection-relaxation evolution. We demonstrate accuracy and performance of the devised scheme for prototypic gas flows over a wide range of rarefaction parameters. Due to the resulting robustness and flexibility of the devised exponential BGK integrator, the scheme paves the way towards more affordable simulations of large-scale and multiscale rarefied gas phenomena.

Figure 1

Pressure tensor and heat flux relaxations with different time step sizes $\mathrm{\Delta }t$ using SP-BGK method. The finely resolved DSMC results are provided as the reference. The time is normalized using the relaxation time $\tau$.

Figure 2

Pressure tensor and heat flux relaxations with different time step sizes $\mathrm{\Delta }t$ using the ED-SP-BGK method with the linear approximation. The finely resolved DSMC results are provided as the reference. The time is normalized using the relaxation time $\tau$.

Figure 3

Pressure tensor and heat flux relaxation with different time step sizes $\mathrm{\Delta }t$ using the ED-SP-BGK method with the exponential approximation. The finely resolved DSMC results are provided as the reference.

Figure 4

Rarefied Sod shock tube test case. ED-SP-BGK, and reference DSMC results were computed using the same spatiotemporal resolution.

Figure 5

Continuum Sod shock tube test case. ED-SP-BGK and reference DSMC results were computed, where much coarser resolution has been employed for ED-SP-BGK.

Figure 6

Comparison of temperature using ED-SP-BGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.1$.

Figure 7

Comparison of velocity $v_y$ using ED-SP-BGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.1$.

Figure 8

Comparison of temperature using ED-SP-BGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.01$.

Figure 9

Comparison of velocity $v_y$ using ED-SP-BGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.01$.

Figure 10

Comparison of temperature using ED-SP-BGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.001$.

Figure 11

Comparison of velocity $v_y$ using ED-SPBGK and SP-BGK as well as different time steps $\mathrm{\Delta }t$ for Couette flow with $\mathrm{Kn}=0.001$.

Figure 12

Comparison of velocity $v_x$ using ED-SP-BGK and SP-BGK as well as different resolutions for Poiseuille flow with $\mathrm{Kn}=0.1$.

Figure 13

Comparison of $L^2$ error using ED-SP-BGK and SP-BGK for Poiseuille flow with $\mathrm{Kn}=0.1$.

Figure 14

Comparison of velocity $v_x$ using ED-SP-BGK and SP-BGK as well as different resolutions for Poiseuille flow with $\mathrm{Kn}=0.001$.

Figure 15

Comparison of $L^2$ error using ED-SPBGK and SPBGK for Poiseuille flow with $\mathrm{Kn}=0.001$.