Symmetrization of advection-diffusion operators

TL;DR

This paper introduces a novel method to convert non-selfadjoint advection-diffusion operators into self-adjoint ones using combined transforms and asymptotic expansion, demonstrated explicitly in shear flow.

Contribution

The paper develops a new transformation technique for advection-diffusion operators, enabling their symmetrization through a combination of point and Lie transforms with asymptotic expansion.

Findings

Exact transformation in shear flow case

Explicit steps for the symmetrization process

Potential for broader application to similar operators

Abstract

We present a new method to transform an expanded class of non-selfadjoint advection-diffusion operators into self-adjoint operators. The transform is based on a combination of a point transform and Lie transform in conjunction with an asymptotic expansion in terms of the diffusivity. We illustrate the method in the context of simple shear flow where the expansion is exact and all transformation steps can be performed explicitly.