Derivation and analysis of Lattice Boltzmann schemes for the linearized Euler equations

Otte, Philipp; Frank, Martin

doi:10.1016/j.camwa.2015.12.004

Cited by 5 publications

(6 citation statements)

References 41 publications

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“…The Lattice Boltzmann equation [7] (LBE) is linear, but actually its nonlinearity is embedded in the left side of the LBE. In LBM, the nonlinear convection term in the macroscopic method is replaced by the linear transfer process, which is similar to the method of solving the compressible flow characteristics.…”

Section: The Lattice Boltzmann Methodsmentioning

confidence: 99%

Flow Simulation of Suspension Bridge Cable Based on Lattice-Boltzmann Method

Huang

Deng

Wang³

2016

Mathematical Problems in Engineering

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Suspension bridge is a kind of bridge which uses cables as the main bearing structure. Suspension bridge has the characteristics of saving materials and weak stiffness. With the increase of the span of suspension bridge, wind induced vibration has resulted in injury of several suspension bridges, which leads to a significant loss. Thus, it is imperative to study the wind vibration mechanism of cables. As for this problem, this paper based on motion theory of mesoscopic particles performs flow simulation of cables by LBM which is different from traditional computing method of fluid mechanics. By calculating the distribution function of the distribution on the grid of uniform flow field, the macroscopic motion law of the flow field around cables can be obtained, which can provide reference for wind resistant design of suspension.

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Section: The Lattice Boltzmann Methodsmentioning

confidence: 99%

Flow Simulation of Suspension Bridge Cable Based on Lattice-Boltzmann Method

Huang

Deng

Wang³

2016

Mathematical Problems in Engineering

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“…Otte [15]) and for the compressible flows (cf. Otte and Frank [16]), this is not true for the LEE with a non-vanishing background velocity. This is particularly due to the additional terms…”

Section: Application To the 3d Linearized Euler Equationsmentioning

confidence: 98%

“…with moment matrix M (1) and equilibrium matrix E (1) which is consistent to α-th order. In addition, assume the relative moment matrix M (2) defined according to (16), the equilibrium matrix E (2) defined according to (17), and a diagonal and positive definite matrix Λ (2) defined analog to (14) fulfilling condition (19). Then, the collision matrix…”

Section: and Ementioning

confidence: 99%

“…In Otte and Frank [16], the authors describe the derivation of stable and second-order consistent LBMs for the compressible Euler equations with zero background velocity for mono-and polyatomic gases. This derivation is based on the discretization of the equilibrium in the acoustic limit presented by Bardos et al [2] and used an a posteriori stability analysis using sufficient conditions for stability in the discrete L 2 -norm also derived in Otte and Frank [16]. The attempts of one of the authors to derive stable LBMs for the isothermal linearized Euler equations with non-vanishing background velocity using the afore mentioned approaches proved not successful.…”

Section: Introductionmentioning

confidence: 99%

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A structured approach to the construction of stable linear Lattice Boltzmann collision operator

Otte

Frank

2020

Computers & Mathematics with Applications

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We introduce a structured approach to the construction of linear BGK-type collision operators ensuring that the resulting Lattice-Boltzmann methods are stable with respect to a weighted L 2 -norm. The results hold for particular boundary conditions including periodic, bounce-back, and bounce-back with flipping of sign boundary conditions. This construction uses the equivalent moment-space definition of BGK-type collision operators and the notion of stability structures as guiding principle for the choice of the equilibrium moments for those moments influencing the error term only but not the order of consistency. The presented structured approach is then applied to the 3D isothermal linearized Euler equations with non-vanishing background velocity. Finally, convergence results in the strong discrete L ∞ -norm highlight the suitability of the structured approach introduced in this manuscript.

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“…These stability structures were evaluated more closely by Junk et al [23], Yong [44], and Rheinländer [37]. Otte et al [32] applied von Neumann analysis [26,40] for the case of a linear equilibrium. Therein, schemes which are found to be linearly stable also obey the stability structures proposed in [37], which validates the novel approach.…”

Section: Introductionmentioning

confidence: 99%

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations

Simonis

Frank

Krause

2020

Phil. Trans. R. Soc. A.

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The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

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Derivation and analysis of Lattice Boltzmann schemes for the linearized Euler equations

Cited by 5 publications

References 41 publications

Flow Simulation of Suspension Bridge Cable Based on Lattice-Boltzmann Method

Flow Simulation of Suspension Bridge Cable Based on Lattice-Boltzmann Method

A structured approach to the construction of stable linear Lattice Boltzmann collision operator

On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations

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