# Nonlinear fourth order Taylor expansion of lattice Boltzmann schemes

## TL;DR

This paper develops a nonlinear fourth-order Taylor expansion for lattice Boltzmann schemes, providing explicit formulas and validation with the D2Q9 scheme to improve understanding of scheme accuracy.

## Contribution

It introduces a nonlinear fourth-order expansion framework for lattice Boltzmann schemes, coupling asymptotic corrections with explicit formulas for conserved and nonconserved moments.

## Key findings

- Validated algebraic expressions with previous results
- Derived explicit formulas for nonlinear fourth-order accuracy
- Applied framework to isothermal D2Q9 lattice Boltzmann scheme

## Abstract

We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinear case up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this approach. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.