中文摘要：

研究了Lorenz非线性系统中使用的集合平均方法来减小计算误差的效果,通过检查5组数值试验（每组20个样本）的结果发现：集合平均对计算误差的减小和消除不如高精度算法有效,这主要体现在以下几方面：1）普通的算法和双精度的计算环境中,若截断误差是主导误差（当初值误差很小时）,各集合的平均结果并不收敛于真值,而是收敛于含截断误差的数值解;2）若初值误差为主导时,系统受到初值误差增长规律的影响,数值解收敛于由初值误差主导的误差解;3）这两种误差量级接近的时候,两种误差都无法消除掉。对解的统计特征进行研究表明,可信的数值解与含计算误差的数值解有许多相似的地方,但是与集合平均的数值解有很大不同,同样说明了集合平均不适用于减小计算误差这样的问题。此外,试验结果表明即使数值解的概率分布形式基本正确,也不能保证数值解是正确的。

英文摘要：

Five groups（each includes 20 samples） of numerical experiments are implemented and the solutions are compared with reliable solutions of the Lorenz chaotic system. Results indicate that the ensemble mean method is not as good as the high precision scheme in reducing the numerical error. 1） When a general scheme and double-precision are used in computing, the truncation error will be dominant, and the ensemble mean solution will not converge to the real solution but approach to a solution with the truncation error. 2） When the initial error is dominant compared to the truncation error, the error will increase exponentially, and the solution will converge to an error-induced solution that is mainly affected by the initial error. 3） When the initial error and the truncation error are comparable, neither of them can be eliminated. The probability density function（PDF） of the numerical solutions is also analyzed for the ensemble mean dynamical system. Results indicate that the PDF is significantly different to that in the original system. This difference further indicates that the ensemble mean method cannot yield a true numerical solution. The PDF study also suggests that a correct PDF distribution of the numerical solution cannot guarantee a correct solution.