Computing effective diffusivity of chaotic and stochastic flows using structure preserving schemes

TL;DR

This paper introduces a structure-preserving numerical scheme for computing the effective diffusivity in chaotic and stochastic flows, addressing challenges in traditional methods and demonstrating improved accuracy and efficiency.

Contribution

A novel stochastic splitting integrator that preserves structure and outperforms Euler methods for solving SDEs in flow problems.

Findings

The new integrator accurately computes effective diffusivity in chaotic flows.

The method effectively captures residual diffusion phenomena.

Numerical results confirm improved performance over standard methods.

Abstract

In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit traditional numerical methods typically fail since the solutions of the advection-diffusion equation develop sharp gradients. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modelled by stochastic differential equations (SDEs). We propose a new numerical integrator based on a stochastic splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem can be viewed as a perturbation. We provide rigorous error analysis for the new numerical integrator using the backward error analysis technique and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow problems of physical interests.