Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9

TL;DR

This paper defines isotropy conditions for lattice Boltzmann schemes using equivalent equations and applies these conditions to optimize the D2Q9 scheme for higher-order isotropy.

Contribution

It introduces a systematic method to select scheme parameters ensuring desired isotropy order in lattice Boltzmann methods.

Findings

Classical constraints for first and second order isotropy in D2Q9.

New non-classical constraints for third and fourth order isotropy.

Method to optimize scheme parameters based on isotropy order.

Abstract

In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d'Humi\'eres. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a lattice Boltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders.