Comparison of different approaches to evaluation of statistical error of the DSMC method

Abstract

Although the direct simulation Monte Carlo (DSMC) method is widely used for solving the steady problems of the rarefied gas dynamics, the questions of its statistical error evaluation are far from being absolutely clear. Typically, the statistical error in the Monte Carlo method is estimated by the standard deviation determined by the variance of the estimate and the number of its realizations. It is assumed that sampled realizations are independent. In distinction from the classical Monte Carlo method, in the DSMC method the time-averaged estimate is used and the sampled realizations are dependent. Additional difficulties in the evaluation of the statistical error are caused by the complexity of the estimates used in the DSMC method. In the presented work we compare two approaches to evaluating the statistical error. One of them is based on the results of the equilibrium statistical mechanics and the "persistent random walk". Another approach is based on the central limit theorem for Markov processes. Each of these approaches has its own benefits and disadvantages. The first approach mentioned above does not require additional computations to construct estimates of the statistical error. On the other hand it allows evaluating statistical error only in the case when all components of velocity and temperature are equivalent. The second approach to evaluating the statistical error is applicable to simulation by the DSMC method a flows with any degree of nonequilibrium. It allows evaluating the statistical errors of the estimates of velocity and temperature components. The comparison of these approaches was realized on the example of a number of classic problems with different degree of nonequilibrium.