Error analysis of an acceleration corrected diffusion approximation of Langevin dynamics with background flow

TL;DR

This paper provides rigorous error estimates for an acceleration corrected diffusion approximation of Langevin dynamics with background flow, demonstrating its accuracy in both short and long-term behaviors.

Contribution

It establishes the first error bounds for this approximation in the averaging regime and confirms its effectiveness through numerical experiments.

Findings

Error in strong sense is O(ε)

Error in weak sense is O(ε^2)

Approximation captures long-time behavior

Abstract

We consider the problem of approximating the Langevin dynamics of inertial particles being transported by a background flow. In particular, we study an acceleration corrected advection-diffusion approximation to the Langevin dynamics, a popular approximation in the study of turbulent transport. We prove error estimates in the averaging regime in which the dimensionless relaxation timescale $\varepsilon$ is the small parameter. We show that for any finite time interval, the approximation error is of order $\mathcal{O}(\varepsilon)$ in the strong sense and $\mathcal{O}(\varepsilon^2)$ in the weak sense, whose optimality is checked against computational experiment. Furthermore, we present numerical evidence suggesting that this approximation also captures the long-time behavior of the Langevin dynamics.