Do current lattice Boltzmann methods for diffusion and diffusion-type equations respect maximum principles and the non-negative constraint?

TL;DR

This paper critically examines whether current lattice Boltzmann methods for diffusion and advection-diffusion equations uphold maximum principles and non-negativity, revealing limitations and proposing improved boundary discretization techniques.

Contribution

The paper demonstrates that existing lattice Boltzmann methods fail to preserve key mathematical properties and introduces a new boundary discretization approach to address this issue.

Findings

Current LBM methods do not preserve maximum principles or non-negativity.

Discretization of boundary conditions significantly impacts LBM performance.

A new boundary discretization method improves adherence to mathematical principles.

Abstract

The lattice Boltzmann method (LBM) has established itself as a valid numerical method in computational fluid dynamics. Recently, multiple-relaxation-time LBM has been proposed to simulate anisotropic advection-diffusion processes. The governing differential equations of advective-diffusive systems are known to satisfy maximum principles, comparison principles, the non-negative constraint, and the decay property. In this paper, it will be shown that current single- and multiple-relaxation-time lattice Boltzmann methods fail to preserve these mathematical properties for transient diffusion-type equations. It will also be shown that the discretization of Dirichlet boundary conditions will affect the performance of lattice Boltzmann methods in meeting these mathematical principles. A new way of discretizing the Dirichlet boundary conditions is also proposed. Several benchmark problems have been solved to illustrate the performance of lattice Boltzmann methods and the effect of discretization of boundary conditions with respect to the aforementioned mathematical properties for transient diffusion and advection-diffusion equations.