Efficient Collision Algorithms in DSMC for Rarefied Gas Dynamics: Markovian NTC-Pre-Scan and Bernoulli-Trial Schemes

The collision process is essential to the Direct Simulation Monte Carlo (DSMC) method, as it incorporates the fundamental principles of the Boltzmann and Kac stochastic equations. A series of collision algorithms, known as the Bernoulli-trials (BT) family schemes, have been proposed based on the Kac stochastic equation. The primary impetus of this paper is to rectify a long-standing theoretical flaw in the widely used no-time-counter (NTC) collision algorithm. We demonstrate that the standard NTC scheme is fundamentally non-Markovian, relying on a fixed majorant product that introduces a system 'memory' and leads to inaccuracies at low particle counts. We propose a new algorithm, NTC-Pre-Scan, which transforms the scheme into a fully Markovian process. When the repeated collisions are not crucial, our new NTC scheme, called NTC-Pre-Scan, could work accurately with a very low number of particles per cell (PPC), average PPC<1, i.e., PPC=0.01, which means with several empty cells in simulations. This contrasts with the standard NTC schemes, which typically require a PPC greater than 1. Then, a systematic evaluation of different BT-based collision partner selection schemes, including the simplified Bernoulli trials (SBT), generalized Bernoulli trials (GBT), symmetrized and simplified Bernoulli trials (SSBT), and the newly proposed symmetrized and generalized Bernoulli trials (SGBT), is conducted to treat some benchmark rarefied gas dynamics problems. The results show that the BT-based collision algorithms and NTC-Pre-scan successfully maintain the collision frequency as the number of particles per cell decreases.