Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor-Green and annular channel flows

TL;DR

This paper evaluates the convergence properties of the Meshless Lattice Boltzmann Method (MLBM) in benchmark fluid flow problems, demonstrating its superior accuracy over traditional LBM and analyzing its error behavior.

Contribution

It provides a detailed convergence analysis of MLBM, compares it with LBM and analytical solutions, and proposes an error expression accounting for the semi-Lagrangian approach.

Findings

MLBM outperforms LBM in error reduction for the same discretization.

LBM errors serve as lower bounds for MLBM errors at the same parameters.

An error expression combining LBM error and semi-Lagrangian terms is proposed.

Abstract

The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in two benchmark tests: the Taylor-Green vortex and annular (bent) channel flow. We compare our MLBM results to LBM and to the analytical solution of the Navier-Stokes equation. We investigate the method's convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance $\delta x$ and timestep length $\delta t$ are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.