  In this paper, we study the convergence analysis for a robust stochastic
structure-preserving Lagrangian numerical scheme in computing effective
diffusivity of time-dependent chaotic flows, which are modeled by stochastic
differential equations (SDEs). Our numerical scheme is based on a splitting
method to solve the corresponding SDEs in which the deterministic subproblem is
discretized using structure-preserving schemes while the random subproblem is
discretized using the Euler-Maruyama scheme. We obtain a sharp and
uniform-in-time convergence analysis for the proposed numerical scheme that
allows us to accurately compute long-time solutions of the SDEs. As such, we
can compute the effective diffusivity for time-dependent flows. Finally, we
present numerical results to demonstrate the accuracy and efficiency of the
proposed method in computing effective diffusivity for the time-dependent
Arnold-Beltrami-Childress (ABC) flow and Kolmogorov flow in three-dimensional
space.
