# Discrete effects on some boundary schemes of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations

## TL;DR

This paper analyzes the discrete effects of boundary schemes in the multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations, providing relations to eliminate numerical slip and extending results to complex-valued equations.

## Contribution

It offers a general analysis of boundary scheme discrete effects, introduces a flexible parameter adjustment method, and extends the analysis to complex-valued convection-diffusion equations.

## Key findings

- Discrete effects are independent of the transform matrix choice.
- Relations are derived to eliminate numerical slip in ABB and BB schemes.
- Numerical examples confirm the effectiveness of the proposed relations.

## Abstract

In this paper, we perform a more general analysis on the discrete effects of some boundary schemes of the popular one- to three-dimensional DnQq multiple-relaxation-time lattice Boltzmann model for convection-diffusion equation (CDE). Investigated boundary schemes include anti-bounce-back(ABB) boundary scheme, bounce-back(BB) boundary scheme and non-equilibrium extrapolation(NEE) boundary scheme. In the analysis, we adopt a transform matrix $\textbf{M}$ constructed by natural moments in the evolution equation, and the result of ABB boundary scheme is consistent with the existing work of orthogonal matrix $\textbf{M}$. We also find that the discrete effect does not rely on the choice of transform matrix, and obtain a relation to determine some of the relaxation-time parameters which can be used to eliminate the numerical slip completely under some assumptions. In this relation, the weight coefficient is considered as an adjustable parameter which makes the parameter adjustment more flexible. The relaxation factors associated with second moments can be used to eliminate the numerical slip of ABB boundary scheme and BB boundary scheme while the numerical slip can not be eliminated of NEE boundary scheme. Furthermore, we extend the relations to complex-valued CDE, several numerical examples are used to test the relations.