On a superconvergent lattice Boltzmann boundary scheme

TL;DR

This paper analyzes the conditions under which lattice Boltzmann boundary schemes achieve superconvergence, revealing that 'magic parameters' depend on moment choices and flow driving methods, impacting boundary accuracy.

Contribution

It provides a detailed analysis of boundary scheme accuracy in lattice Boltzmann methods, showing the dependence of 'magic parameters' on moments and flow conditions.

Findings

'Magic parameters' depend on moment choices.

Boundary accuracy is influenced by flow driving methods.

Superconvergence conditions vary with scheme details.

Abstract

In a seminal paper Ginzburg and Adler analyzed the bounce-back boundary conditions for the lattice Boltzmann scheme and showed that it could be made exact to second order for the Poiseuille flow if some expressions depending upon the parameters of the method were satisfied, thus defining so-called "magic parameters". Using the Taylor expansion method that one of us developed, we analyze a series of simple situations (1D and 2D) for diffusion and for linear fluid problems using bounce-back and "anti bounce-back" numerical boundary conditions. The result is that "magic parameters" depend upon the detailed choice of the moments and of their equilibrium values. They may also depend upon the way the flow is driven.