Study on the Ensemble Mean Method in Reducing the Numerical Error in Chaotic Dynamical System

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.