A structure-preserving scheme for computing effective diffusivity and anomalous diffusion phenomena of random flows

TL;DR

This paper develops a structure-preserving numerical scheme to accurately compute effective diffusivity and analyze anomalous diffusion in stochastic flows, demonstrating improved convergence and error estimates over traditional methods.

Contribution

It introduces a novel one-step structure-preserving method for solving SDEs related to particle diffusion in random flows, with rigorous convergence and error analysis.

Findings

The scheme accurately computes effective diffusivity in 2D and 3D random fields.

It captures the power law relationship in anomalous diffusion phenomena.

The method outperforms Euler-Maruyama in convergence and error estimates.

Abstract

This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields.