A fast algorithm for the estimation of statistical error in DNS (or experimental) time averages

Time- and space-averaging of the instantaneous results of DNS (or experimental measurements) represent a standard final step, necessary for the estimation of their means or correlations or other statistical properties. These averages are necessarily performed over a finite time and space window, and are therefore more correctly just estimates of the ‘true’ statistical averages. The choice of the appropriate window size is most often subjectively based on individual experience, but as subtler statistics enter the focus of investigation, an objective criterion becomes desirable. Here a modification of the classical estimator of averaging error of finite time series, i.e. ‘batch means’ algorithm, will be presented, which retains its speed while removing its biasing error. As a side benefit, an automatic determination of batch size is also included. Examples will be given involving both an artificial time series of known statistics and an actual DNS of turbulence.