Entropic lattice Boltzmann method for general anisotropic advection--diffusion

TL;DR

This paper introduces a local entropic lattice Boltzmann method for accurately simulating general anisotropic advection--diffusion processes, even with high tensor contrasts and rotated principal axes.

Contribution

It develops a novel discretization scheme that handles rotated, heterogeneous, and dynamically coupled anisotropic tensors with high accuracy and stability.

Findings

Validated on 3D benchmarks including Gaussian plumes and Fourier modes.

Achieved accurate simulations with anisotropy ratios up to 10^4.

Applied to Taylor dispersion and Rayleigh--Bénard convection, demonstrating robustness.

Abstract

Many transport processes exhibit direction-dependent diffusion, described macroscopically by the full-tensor anisotropic advection--diffusion equation (ADE). Numerical discretization is demanding when the principal axes are rotated relative to the mesh, since mixed derivatives and oblique fluxes amplify grid-orientation errors under large tensor contrasts. This paper develops a local entropic lattice Boltzmann discretization for the general anisotropic ADE. The non-equilibrium population is split into a first-order flux sector and a residual ghost sector. The diffusion tensor is imposed through local tensorial relaxation of the flux, while higher-order kinetic content is controlled by an ADE-corrected entropic stabilizer with positivity fallback. Chapman--Enskog analysis shows the scheme recovers the target full-tensor equation with a discrete-time diffusivity relation between the physical tensor and the flux-relaxation matrix. The update is local, matrix-free, and applies to rotated, spatially varying, heterogeneous, and dynamically coupled tensor transport. We validate it on 3D benchmarks--advected Gaussian plumes, decay of rotated Fourier modes, and source-driven transport with varying tensors--covering off-diagonal diffusion, high-P\'eclet advection, anisotropy ratios of O(104)O(10^4) O(104), and local contrasts up to $3\times10^4:1$. It is then applied to orientation-induced Taylor dispersion of Brownian rods, quantifying enhancement from shear-driven rotation. Heat-conduction tests include rotated thermal-conductivity measurements and effective conduction in heterogeneous porous media with anisotropy up to $10^4:1. Finally, anisotropic Rayleigh--B\'enard convection is simulated to examine how plume morphology and heat transfer change over seven decades of anisotropy ratios, demonstrating an accurate, stable local solver for strongly anisotropic advection--diffusion.