# Recursive regularization step for high-order lattice Boltzmann methods

## TL;DR

This paper introduces a recursive regularization step for high-order lattice Boltzmann methods, significantly improving their stability and accuracy in simulating complex fluid flows, especially under high Reynolds numbers.

## Contribution

The paper develops a recursive computation of Hermite coefficients for LBM, reducing computational cost and enhancing stability and accuracy over standard regularization methods.

## Key findings

- Enhanced stability range demonstrated in shear layer simulations at high Reynolds numbers.
- Recursive regularization outperforms standard methods in thermal and compressible flow simulations.
- Confirmed stability improvements through Sod shock tube tests with high-order lattices.

## Abstract

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of non-equilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second (and higher) order non-hydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from $10^4$ to $10^6$, and where a thorough analysis of the case at $Re=3\times 10^4$ is conducted. In the latter, results obtained using both regularization steps are compared against the BGK-LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.