On single distribution lattice Boltzmann schemes for the approximation of Navier Stokes equations

TL;DR

This paper investigates the capability of single distribution lattice Boltzmann schemes to accurately approximate Navier-Stokes equations, analyzing algebraic conditions for equilibrium functions and relaxation parameters across various lattice configurations.

Contribution

It provides a detailed algebraic analysis of 12 classical lattice Boltzmann schemes, proposing new equilibrium expressions to better approximate Navier-Stokes equations.

Findings

Second-order accuracy in bidimensional and three-dimensional schemes

Identification of cases where exact physical model fitting is impossible

Adjusted equilibrium functions for improved approximation

Abstract

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.