Coupling the molecular motion and collision processes in numerical simulations

Abstract

The molecular motion and collision processes are usually decoupled in the traditional molecular simulations, where the simulation process is divided into a series of time steps and the two processes are sequentially executed during each time step. The numerical errors in transport properties and flow-field solutions will become noticeable when the time step is much larger than the mean time interval between intermolecular collisions. The limitation of using small time step can be relaxed for multiscale problems by using coupled algorithm that allows the molecular motions and collisions to happen simultaneously. This coupling idea was proposed in the DSBGK method that however focused on the discussion of variance reduction. The same coupling idea has been also implemented in the recent USP-ESBGK method that focused on the advantage of using a coupled algorithm. As this is a significant advancement in particle simulation, we present the similarity analysis between the two methods in the coupling spirit as well as the difference in the detailed implementations.

Introduction

In the traditional molecular simulation methods, including the deterministic molecular dynamics (MD) method [1] and the direction simulation Monte Carlo (DSMC) method [2], [3] among others, the simulation process is divided into a series of time steps; during each time step, the molecular motion and collision processes are decoupled and executed as two sequential steps. Consequently, when the time step is much larger than the mean time interval between intermolecular collisions, the molecular motion process will skip the cells located along its trajectory except the last cell where the molecule concerned is situated at the end of the current time step. In the subsequent intermolecular collision process, each molecule will have collisions with other molecules located inside the same last cell and thus will contribute only to the last cell. However, if the time step is larger than the molecular mean collision time, each molecule in reality has noticeable chance to have collisions with the encountered molecules during its pass, which should also change the solutions at the otherwise skipped cells in addition to the last cell. Therefore, to make the use of large time step valid, the intermolecular collision process should be coupled with the molecular motion process such that the collision effect is truthfully reflected in the solutions of all cells located along the molecular trajectory. In the MD simulations, a cell normally depends on the intermolecular forces and the position of molecule concerned, while in the DSMC simulations cells are preset in the initialization and used to localize the intermolecular collisions.

The unified stochastic particle method based on ESBGK model (USP-ESBGK) [4] was recently proposed to couple the molecular motion and collision processes in numerical simulations. Actually, this coupling idea has been previously proposed in the direct simulation BGK (DSBGK) method [5], which however focused on its noticeable advantage in variance reduction in simulating low-speed flow problems with low signal-to-noise ratio (e.g., Mach number is smaller than ${10}^{-4}$). A coupled algorithm will have broad applications and significant impacts in reducing computational cost. Thus, it is very important to clarify the original contribution by each group. In the following Section 2, we present the coupling idea applied in the two molecular simulation methods. The complete DSBGK method is also introduced in Section 3 to show its simplicity and elegance in achieving this coupling idea.