Finite Difference formulation of any lattice Boltzmann scheme

TL;DR

This paper presents a rigorous framework that reformulates any lattice Boltzmann scheme as a finite difference scheme, enabling precise analysis of consistency and stability using classical finite difference tools.

Contribution

It introduces a formalism to rewrite lattice Boltzmann schemes as finite difference schemes, facilitating rigorous analysis of their stability and consistency.

Findings

Reformulation of lattice Boltzmann schemes as finite difference schemes

Alignment of stability analysis with classical von Neumann analysis

Numerical illustrations confirming the theoretical framework

Abstract

Lattice Boltzmann schemes rely on the enlargement of the size of the target problem in order to solve PDEs in a highly parallelizable and efficient kinetic-like fashion, split into a collision and a stream phase. This structure, despite the well-known advantages from a computational standpoint, is not suitable to construct a rigorous notion of consistency with respect to the target equations and to provide a precise notion of stability. In order to alleviate these shortages and introduce a rigorous framework, we demonstrate that any lattice Boltzmann scheme can be rewritten as a corresponding multi-step Finite Difference scheme on the conserved variables. This is achieved by devising a suitable formalism based on operators, commutative algebra and polynomials. Therefore, the notion of consistency of the corresponding Finite Difference scheme allows to invoke the Lax-Richtmyer theorem in the case of linear lattice Boltzmann schemes. Moreover, we show that the frequently-used von Neumann-like stability analysis for lattice Boltzmann schemes entirely corresponds to the von Neumann stability analysis of their Finite Difference counterpart. More generally, the usual tools for the analysis of Finite Difference schemes are now readily available to study lattice Boltzmann schemes. Their relevance is verified by means of numerical illustrations.