Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

TL;DR

This paper develops a methodology to analyze the hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes, clarifying how different collision models affect accuracy and stability in low-viscosity flows.

Contribution

It introduces a systematic approach to derive and validate hydrodynamic limits and error terms for various LBM collision models, enhancing understanding of their numerical properties.

Findings

Low dissipation in BGK linked to specific error patterns.

Regularized and MRT models exhibit hyperviscous degeneracy.

Error analysis explains over-dissipation and instabilities.

Abstract

With the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical pre-factor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime.