The incompressible Navier-Stokes limit from the lattice BGK Boltzmann equation

TL;DR

This paper proves the derivation of the incompressible Navier-Stokes equations from a velocity-discretized Boltzmann equation with a simplified BGK collision operator, including numerical insights on the convergence rate.

Contribution

It establishes the hydrodynamic limit from a discretized Boltzmann equation to Navier-Stokes equations and characterizes velocity configurations that lead to this limit.

Findings

Hydrodynamic limit can be achieved from the discretized Boltzmann equation.

Numerical computations in 2D quantify the convergence rate as Knudsen number approaches zero.

Characterization of particle velocities and probabilities leading to Navier-Stokes equations.

Abstract

In this paper, we prove that a local weak solution to the $d$-dimensional incompressible Navier-Stokes equations ($d \geq 2$) can be constructed by taking the hydrodynamic limit of a velocity-discretized Boltzmann equation with a simplified BGK collision operator. Moreover, in the case when the dimension is $d=2,3$, we characterize the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier-Stokes equations in the hydrodynamic limit. Numerical computations conducted in 2D provide information about the rate with which this hydrodynamic limit is achieved when the Knudsen number tends to zero.