Equivalent finite difference and partial differential equations for the lattice Boltzmann method

Fučík, Radek; Straka, Róbert

doi:10.1016/j.camwa.2021.03.014

“…The difficulty in establishing a general result comes from the coupling between spatial operators and time shifts. We must mention that during the drafting of the present contribution, an interesting work by Fucik and Straka [21] has been published covering the very same subject and essentially coming to the same conclusion as our paper. Their focus is different than ours since they adopt a purely algorithmic approach rather than a precise algebraic characterization of lattice Boltzmann schemes.…”

Section: Introductionsupporting

confidence: 65%

“…There are cases, which seem quite rare according to our experience (we succeeded in finding only one special case where this happens), where the answer is positive. This phenomenon has also been discussed by [21], without envisioning a systematic way of guaranteeing the minimality of the Finite Difference scheme obtained by their algorithm.…”

Section: Examples Simplifications and Particular Casesmentioning

confidence: 99%

“…Consider the D 1 Q 3 SRT scheme by [21], also corresponding to that of [39] which reads with our notations d = 1, q = 3 and c 1 = 0, c 2 = 1 and c 3 = −1 and…”

Section: Appendicesmentioning

confidence: 99%

“…Consider the D 1 Q 3 MRT scheme by [21], which is constructed in the same way than the previous one except for S = diag(0, ω 2 , ω 3 ) with ω 2 , ω 3 = 1. The characteristic and minimal polynomial coincide and are given by…”

Section: Appendicesmentioning

confidence: 99%

“…We actually provide more insight into the bound on the number of time steps of the corresponding Finite Difference scheme and our formalism, based on polynomials, aims at providing a direct link with the classical tools for the stability analysis and allows to establish a link with the Taylor expansions from [17], as introduced in [1]. In [21], the authors rely on a decomposition of the scheme using an hollow matrix 2 yielding an equivalent form of the scheme with the diagonal non-equilibrium part, after a finite number of steps of their algorithm. However, to the best of our understanding, the origin of such algorithm is not fully clear.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Finite Difference formulation of any lattice Boltzmann scheme

Bellotti¹,

Graille²,

Massot³

2022

Preprint

1

11

0

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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.

show abstract

Finite Difference formulation of any lattice Boltzmann scheme

Bellotti

¹

,

Graille

²

,

Massot

³

2022

View full text Add to dashboard Cite

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.

show abstract