Third order equivalent equation for the relative velocity lattice Boltzmann schemes with one conservation law

TL;DR

This paper derives the third order equivalent equation for relative velocity lattice Boltzmann schemes with one conservation law, analyzing their formal precision and asymptotics across multiple dimensions and velocities.

Contribution

It introduces a general third order equivalent equation for these schemes, extending understanding of their asymptotic behavior in various dimensions.

Findings

Derived the third order equivalent equation for the schemes.

Analyzed the asymptotics for arbitrary dimensions and velocities.

Clarified the formal precision of the relative velocity lattice Boltzmann schemes.

Abstract

We study the formal precision of the relative velocity lattice Boltzmann schemes. They differ from the d'Humi\`eres schemes by their relaxation phase: it occurs for a set of moments parametrized by a velocity field function of space and time. We deal with the asymptotics of the relative velocity schemes for one conservation law: the third order equivalent equation is exposed for an arbitrary number of dimensions and velocities.