Convergence analysis of the direct simulation Monte Carlo based on the physical laws of conservation

Abstract

Computational errors in the direct simulation Monte Carlo method can be categorized into four types; decomposition (or discretization), statistical, machine, and boundary condition errors. They arise due to variety of reasons including decoupling of movement and collision phases into two separate steps, finiteness of molecule numbers and domain cell-size, existence of statistical fluctuations and uncertainty, using machines to solve physical problems numerically, computational implementation of boundary conditions of approximate nature, and, finally, assumptions and simplifications adopted in the inter-molecular collision models. In this study, a verification method based on the physical laws of conservation, which are an exact consequence of the Boltzmann equation, is introduced in order to quantify the errors of the DSMC method. A convergence history according to the new verification method is then presented that can illustrate the effects of all type of errors during the simulation run. Convergence analysis indicates that the DSMC method can satisfy the conservation laws with an acceptable level of precision for the flow problems studied. Finally, it is shown that the overall deviation from conservation laws increases with decreasing sample size value and number of particles, and with increasing length of cells and time-step interval size.