Fast-Converging and Asymptotic-Preserving DSMC

TL;DR

This work addresses the slow convergence of DSMC in near-continuum flows by analyzing the direct intermittent general synthetic iterative scheme (DIG) within a linearized BGK framework. It combines a fast-converging macroscopic synthetic update with intermittent DSMC steps, and demonstrates both fast convergence (via Fourier stability) and asymptotic preservation of the Navier–Stokes limit (via Chapman–Enskog expansion), allowing $O(1)$-sized cells. The results show that CIS slows as Knudsen number shrinks, GSIS accelerates convergence, and DIG delivers orders-of-magnitude speedups in near-continuum regimes, while preserving stochastic DSMC behavior. Numerical tests on Planar Poiseuille flow and shock interactions with a cylinder confirm AP-NS behavior and substantial computational savings, suggesting broad engineering applicability, including multicomponent and reacting flows.

Abstract

Improving the efficiency of the direct simulation Monte Carlo (DSMC) method has become increasingly urgent with the rapid development of space exploration. To address this issue, the direct intermittent general synthetic iteration (DIG) scheme has recently been proposed to enable DSMC's rapid and accurate convergence to steady-state solutions, even when the cell size is much larger than the mean free path in near-continuum flow regimes. The first part of the paper is devoted to the mathematical analysis of DIG's fast-converging and asymptotic-preserving properties. Because the Boltzmann equation is analytically intractable, the analysis is conducted using the linearized BGK model. It is found that, in the near continuum flow regime, the DIG method asymptotically recovers the Navier Stokes equations when the cell size is O(1), rather than being constrained by the mean free path. Moreover, after a single cycle of DIG evolution, the deviation from the final steady state solution is reduced by more than a factor of five. In the second part of the paper, the Poiseuille flow and hypersonic flow passing over cylinder are investigated using the DIG scheme, with different time step and cell sizes, thereby demonstrating its efficiency and accuracy in multiscale flow simulation. Specifically, when the Knudsen number is 0.01, the DIG method is found to be faster than the traditional DSMC method by two orders of magnitude. The performance gain becomes even greater at smaller Knudsen numbers. The proposed method holds great potential for engineering applications.

Figures (7)

• Figure 1: Error decay rates as functions of the Knudsen number and time step for CIS and GSIS.
• Figure 1: Velocity profiles in the force-driven Poiseuille flow. H500 and H40 mean that the computational domain is discretized uniformly by 500 and 40 cells, while N20 represents the nonuniform grids generated by \ref{['hyperbolic_tangent_eq']}. A CFL number of 0.2 is applied, based on the minimum grid size.
• Figure 2: Comparison of error decay rates between the CIS and DIG, when $m=100$. In CIS, the time step is $\delta^{-1}_{rp}$, mimicking that in DSMC. In DIG, the time step are $\delta^{-1}_{rp}$ and $\sqrt{\delta^{-1}_{rp}}$, respectively. Note that the error decay rates in CIS and DIG are calculated i a unit DIG cycle, i.e., $|e_c(\Delta t)|^{100}$ and $|e_c(\Delta t)|^{99}\times{}|e_d(\Delta t)|$, respectively.
• Figure 2: Convergence history of velocity profiles at Kn = 0.01. 500 uniform and 20 non-uniform cells are used in DSMC and DIG, respectively. The CFL number in DSMC is 0.2, while that in DIG is 0.2 and 4. Therefore, the time step in the left two figures is the same. For each line, the result is obtained by averaging over the previous 100 time steps.
• Figure 3: First row: computational grids in DSMC (upper half) and DIG (lower half) near the square cylinder. In the open-source DSMC code SPARTA, DSMC requires about 838,545 cells at Kn = 0.1 and 5,814,370 cells at Kn = 0.01, with only 12,750 and 51,000 cells for DIG, respectively. Second row: comparison of the mean free path and grid size between DIG and DSMC. The mean free path first increases and then decreases when approaching the left side of the cylinder because, due to strong compression, the temperature rises first, but is followed by a significant increase in density. The Knudsen numbers are 0.1 and 0.01 in the left and right columns, respectively.