Smooth initialization of lattice Boltzmann schemes

Leemput, Pieter Van; Rheinländer, Martin; Junk, Michael

doi:10.1016/j.camwa.2009.02.022

Cited by 11 publications

(13 citation statements)

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“…It is worthwhile observing that the choice of initial datum does not play any role in the previous procedure and does not influence the stability analysis of Section 6. It only comes into play during the consistency analysis of the numerical method, which is not investigated in this paper, in particular, as far as time boundary layers are concerned, see [40,34].…”

Section: Initialization Schemesmentioning

confidence: 99%

Finite Difference formulation of any lattice Boltzmann scheme

Bellotti¹,

Graille²,

Massot³

2022

Preprint

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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.

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Section: Initialization Schemesmentioning

confidence: 99%

Finite Difference formulation of any lattice Boltzmann scheme

Bellotti¹,

Graille²,

Massot³

2022

Preprint

View full text Add to dashboard Cite

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“…These variables are considered in normed vector spaces U, V and W , in which W is the direct sum of the normed spaces U and V . This is discussed for a lattice Boltzmann scheme in [14] with two distribution functions f −1 and f 1 .…”

Section: Smooth Initializationmentioning

confidence: 99%

“…This makes the idea of this method quite similar to that of the CR-algorithm. To put the idea of smooth initialization into practice, an mregularity condition is defined on output w in [14]. The rule to determine the value v(0) is then given in [14] by: The initial value w(0) of some m-regular output w must coincide with the polynomial backward extrapolation of its values pertaining to the iterations t 1 = ∆t,t 2 = 2∆t, .…”

Section: Smooth Initializationmentioning

confidence: 99%

“…Formulating the lifting operator as a missing data problem is an alternative way to arrive at an implicit expression for the unknown distribution function f (x, v,t). A recent technique to fill in missing data is smooth initialization [14]. In a missing data problem, variable w ∈ W is split into w = (u, v), where u ∈ U contains the explicitly given data and v ∈ V the missing data.…”

Section: Introductionmentioning

confidence: 99%

“…These variables are considered in normed vector spaces U, V and W , in which W is the direct sum of the normed spaces U and V . In [14] this method is used to create a lifting operator for a lattice Boltzmann scheme with two possible velocity directions in a one-dimensional domain.…”

Section: Introductionmentioning

confidence: 99%

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Lifting in hybrid lattice Boltzmann and PDE models

Vanderhoydonc

Vanroose

2011

Comput. Visual Sci.

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Mathematical models based on kinetic equations are ubiquitous in the modeling of granular media, population dynamics of biological colonies, chemical reactions and many other scientific problems. These individual-based models are computationally very expensive because the evolution takes place in the phase space. Hybrid simulations can bring down this computational cost by replacing locally in the domain -in the regions where it is justified -the kinetic model with a more macroscopic description. This splits the computational domain into subdomains. The question is how to couple these models in a mathematically correct way with a lifting operator that maps the variables of the macroscopic partial differential equation to those of the kinetic model. Indeed, a kinetic model has typically more variables than a model based on a macroscopic partial differential equation and at each interface we need the missing data. In this contribution we report on different lifting operators for a hybrid simulation that combines a lattice Boltzmann model -a special discretization of the Boltzmann equation -with a diffusion partial differential equation. We focus on the numerical comparison of various lifting strategies.

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