# Lattice Boltzmann inverse kinetic approach for the incompressible Navier-Stokes equations

## TL;DR

This paper develops a novel inverse kinetic theory for lattice Boltzmann methods that exactly reproduces the incompressible Navier-Stokes equations without asymptotic assumptions, allowing precise control over accuracy.

## Contribution

It introduces an inverse kinetic approach that yields exact fluid equations for arbitrary smooth distributions, surpassing previous entropic LB methods by removing functional constraints.

## Key findings

- Exact kinetic theory for incompressible Navier-Stokes equations
- Asymptotic accuracy estimates for LB methods
- Comparison with Chorin artificial compressibility method

## Abstract

In spite of the large number of papers appeared in the past which are devoted to the lattice Boltzmann (LB) methods, basic aspects of the theory still remain unchallenged. An unsolved theoretical issue is related to the construction of a discrete kinetic theory which yields \textit{exactly} the fluid equations, i.e., is non-asymptotic (here denoted as \textit{LB inverse kinetic theory}). The purpose of this paper is theoretical and aims at developing an inverse kinetic approach of this type. In principle infinite solutions exist to this problem but the freedom can be exploited in order to meet important requirements. In particular, the discrete kinetic theory can be defined so that it yields exactly the fluid equation also for arbitrary non-equilibrium (but suitably smooth) kinetic distribution functions and arbitrarily close to the boundary of the fluid domain. Unlike previous entropic LB methods the theorem can be obtained without functional constraints on the class of the initial distribution functions. Possible realizations of the theory and asymptotic approximations are provided which permit to determine the fluid equations \textit{with prescribed accuracy.} As a result, asymptotic accuracy estimates of customary LB approaches and comparisons with the Chorin artificial compressibility method are discussed.