Numerical approximation of effective diffusivities in homogenization of nondivergence-form equations with large drift by a Lagrangian method

TL;DR

This paper introduces a Lagrangian numerical scheme for efficiently approximating effective diffusivities in homogenized nondivergence-form elliptic equations with large drift, avoiding complex PDE solutions.

Contribution

The paper develops a novel Lagrangian method based on stochastic simulation and modified equations, achieving higher-order convergence without solving Fokker-Planck equations.

Findings

Accurately computes effective diffusivities in high dimensions.

Achieves higher-order weak convergence rates.

Demonstrates efficiency and accuracy through numerical experiments.

Abstract

In this paper, we study numerical methods for the homogenization of linear second-order elliptic equations in nondivergence-form with periodic diffusion coefficients and large drift terms. Upon noting that the effective diffusion matrix can be characterized through the long-time variance of an associated diffusion process, we construct a Lagrangian numerical scheme based on a direct simulation of the underlying stochastic differential equation and utilizing the framework of modified equations, thereby avoiding the need to solve the Fokker--Planck--Kolmogorov equation. Through modified equation analysis, we derive higher-order weak convergence rates for our method. Finally, we conduct numerical experiments to demonstrate the accuracy of the proposed method. The results show that the method efficiently computes effective diffusivities, even in high dimensions.