考虑时间相关性的DSMC方法统计误差分析（英文）

Statistical Error Analysis of the DSMC Method Considering Time Correlations Between Samples

过去的半个世纪,DSMC方法已经成为求解稀薄气体流动最有力的数值工具.它的精度问题得到了许多研究者的关注.DSMC方法的误差可以分为两类,一类是统计误差,另一类是数值误差.在DSMC方法中,宏观流动性质是通过对微观分布信息的抽样平均而得到的,因此模拟结果本质上是符合概率统计的,需要对由于有限的抽样步数而导致的统计误差进行仔细地量化考察.统计误差在DSMC方法中占有重要的地位,但是直到今天仍没有被完全研究清楚.尽管前人做了一些研究,但是通常都假设抽样结果是相互独立的,对于获得精确结果所需的最小抽样步数,还没有清晰统一的认识.时间相关性会增加统计误差理论分析的难度,因此在已有的工作中很少被考虑到.使用统计学中的自相关函数以及修正的中心极限定理,可以量化抽样结果中的时间相关性.在考虑抽样结果时间相关性的基础上,研究了DSMC方法中的统计误差,考核算例为一维的Couette流动问题.量化的结果显示,时间相关性对统计误差的影响很大.时间相关性可增大随机变量抽样序列的方差.

The DSMC method has evolved into a most powerful numerical tool for rarefied gas flow in the past half century. The problems related to accuracy have got much attention in DSMC professions. There are 2 types of errors in the DSMC method. One is termed ＂statistical error＂, and the other is ＂numerical error＂. In the DSMC method, the macroscopic properties are obtained with the sample average of the microscopic information. The simulation results are therefore inherently statistical and statistical errors due to finite sampling need to be fully quan- tiffed. Statistical error plays an important role in the DSMC method. However, it has not been well understood as yet. Most of the investigations are based upon the assumption that the suc- cessive sample results are independent. It is still not clear how many sampling steps are re- quired to get accurate results. Obviously the time correlations make the theoretical analysis of the statistical error more difficult, which has seldom been taken into accounted in the previous researches. With the autocorrelation function and the modified central limit theorem in the sta- tistics, the time correlations between samples can be quantified. The statistical error of the DSMC method was studied based on the benchmark problem of the Couette flow, in view of the time correlations between samples. Quantitative results show that the time correlations affect the statistical error greatly. The time correlations tend to increase the variance of sampled val- ues of random variables, and it takes almost 100 sample steps for the autocorrelation ftmction to decay to 0.