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.

Rigorous derivation of the macroscopic equations for the lattice Boltzmann method via the corresponding Finite Difference scheme

“…Macroscopic equations (N <q continuous PDEs) [3] Small ∆t /∆x/ [8, 37,31] [13, 15,46] Quasi-equilibrium [8,37,31] [13, 15,46] Small ∆t /∆x [41,1] Figure 1: Different paths to recover the macroscopic equations. The formal approaches available in the literature [8,37,31,13,15,46] rely on the path marked with dashed arrows.…”

“…Our way of proceeding is marked with full arrows: we eliminate exactly the non-conserved moments at the discrete level as in [3] and we perform the usual consistency analysis for Finite Difference schemes as in [41,1].…”

“…In this contribution, we develop the other path, namely the top-down movement followed by the left-right one on Figure 1. In particular, in order to fill the hollow between lattice Boltzmann schemes and traditional approaches known to numerical analysts, such as Finite Difference schemes, we recently introduced [3] a formalism to recast any lattice Boltzmann scheme, regardless of its linearity, as a multistep Finite Difference scheme solely on the conserved moments. It should be stressed that our standpoint, where lattice Boltzmann schemes are studied in terms of their Finite Difference counterpart, must not be seen as the right way of implementing them, because one would lose most of the previously mentioned computational efficiency coming from the kinetic vision.…”

“…The price to pay for the non-conserved moments relaxing away from the equilibria is the multi-step nature of the Finite Difference scheme. In our previous proposal [3], it has been crucial to be able to provide, thanks to a systematic mathematical approach, a precise description of the main ingredient needed to reduce the lattice Boltzmann scheme to a Finite Difference scheme, namely the characteristic polynomial of matrices of Finite Difference operators. We are therefore allowed to utilize this characteristic polynomial as a tool satisfying certain properties alone from the particular underlying lattice Boltzmann scheme.…”

“…In Section 2, we set notations and assumptions concerning the lattice Boltzmann schemes we shall work with. Section 3 is devoted to recall the main results from our previous work [3] concerning the recast of any lattice Boltzmann scheme as a Finite Difference scheme. These results are then stated in a slightly different manner, facilitating the following analysis.…”