Analysis of simplifications of numerical schemes for Langevin equation, effect of variations in the correlation of augmentations

The possibility to simplify the integration of Langevin equation using the variation of correlation between augmentation was researched. The analytical expression for a set of numerical schemes is presented. Its shown that asymptotic limits for squared velocity depend on step size. The region of convergence and the convergence orders were estimated. It turned out that the incorrect correlation between increments decrease the accuracy down to the level of first-order methods for schemes based on precise solution.