Finite Difference formulation of any lattice Boltzmann scheme

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 frequentlyused 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.

Limit Consistency of Lattice Boltzmann Equations

“…Parametrizing the latter in advance allows to predetermine the amalgamation of discrete and relaxation parameters, which is primarily relevant to uphold the path towards the targeted PDE. The unfolding of LBM as a chain of finite differences and Taylor expansions under the relaxation constraint matches with results in the literature [16,5] and thus validates the present work. Although, similarities to wellestablished references [17,18] are present, it is to be stressed that the purpose of limit consistency is to touch base with the previous construction steps [33] and enable a generic top-down design of LBM.…”

“…In addition, we couple the discretization to a sequencing parameter which is responsible for the weak convergence of mesoscopic solutions to solutions of the macroscopic target PDE. This finding aligns with recent works [5,4] which rigorously express lattice Boltzmann methods in terms of finite differences for the macroscopic variables.…”

“…As a direct result, we unfold the discretization technique of lattice Boltzmann methods as chaining finite differences and provide a generic top-down derivation of the numerical scheme. Validating our approach, the finite difference interpretation of LBM matches the previous [16] and current rigorous observations [5,4] in the literature.…”

Truncation errors and modified equations for the lattice Boltzmann methodviathe corresponding Finite Difference schemes

“…(N <q continuous PDEs) Modified equations [3] Small ∆t /∆x/ [10, 43,36] [15, 17,53] Quasi-equilibrium [10,43,36] [15, 17,53] Small ∆t /∆x [48, 1,52,8] Figure 1: Different paths to recover the macroscopic equations and the modified equations. The formal approaches available in the literature [10,43,36,15,17,53] rely on the path marked with dashed arrows.…”

“…They perform Taylor expansions for small discretization parameters and then utilize the quasi-equilibrium of the non-conserved moments to get rid of them. Our way of proceeding is marked with full arrows: we eliminate exactly the nonconserved moments at the discrete level as in [3] and we perform the usual analyses for Finite Difference schemes as in [48,1,52,8].…”