Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

TL;DR

This paper presents a convergence analysis of a Lagrangian numerical scheme for accurately computing the effective diffusivity in 3D time-dependent chaotic flows modeled by SDEs, with demonstrated numerical efficiency.

Contribution

It introduces a structure-preserving splitting scheme with uniform-in-time convergence analysis for effective diffusivity computation in complex flows.

Findings

The scheme accurately computes long-time solutions of SDEs.

Numerical results show high accuracy for ABC and Kolmogorov flows.

The method is efficient for 3D time-dependent flow simulations.

Abstract

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.