A multiple-relaxation-time lattice Boltzmann model based four-level finite-difference scheme for one-dimensional diffusion equation

TL;DR

This paper develops a stable, high-accuracy four-level finite-difference scheme derived from a multiple-relaxation-time lattice Boltzmann model for solving the one-dimensional diffusion equation, validated by numerical tests.

Contribution

It introduces a novel explicit four-level finite-difference scheme with sixth-order spatial accuracy derived from an MRT-LB model for 1D diffusion.

Findings

The scheme is unconditionally stable.

It achieves sixth-order spatial accuracy.

Numerical results confirm theoretical predictions.

Abstract

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient $\omega_{0}$ and the relaxation parameters $s_1$ and $s_2$ corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations, and find that the numerical results are consistent with our theoretical analysis.