# Convergence of stochastic structure-preserving schemes for computing effective diffusivity in random flows

## TL;DR

This paper introduces stochastic structure-preserving numerical schemes for accurately computing effective diffusivity in particles moving through random flows, with rigorous convergence analysis and numerical validation.

## Contribution

The paper develops novel stochastic structure-preserving schemes for SDEs in random flows, providing a rigorous convergence analysis and connecting discrete and continuous corrector problems.

## Key findings

- Schemes accurately compute effective diffusivity in random flows.
- Convergence analysis uses probabilistic approach and Markov process interpretation.
- Numerical results demonstrate efficiency and reveal convection-enhanced diffusion phenomena.

## Abstract

In this paper, we propose stochastic structure-preserving schemes to compute the effective diffusivity for particles moving in random flows. We first introduce the motion of particles using the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We also discuss the definition of the corrector problem and effective diffusivity. Then we propose stochastic structure-preserving schemes to solve the SDEs and provide a sharp convergence analysis for the numerical schemes in computing effective diffusivity. The convergence analysis follows a probabilistic approach, which interprets the solution process generated by our numerical schemes as a Markov process. By using the central limit theorem for the solution process, we obtain the convergence analysis of our method in computing long time solutions. Most importantly our convergence analysis reveals the connection of discrete-type and continuous-type corrector problems, which is fundamental and interesting. We present numerical results to demonstrate the accuracy and efficiency of the proposed method and investigate the convection-enhanced diffusion phenomenon in two- and three-dimensional incompressible random flows.