Analysis of the lattice Boltzmann treatment of hydrodynamics

McNamara, Guy R.; Alder, B. J.

doi:10.1016/0378-4371(93)90356-9

Cited by 153 publications

(90 citation statements)

References 10 publications

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“…This approach has much in common with explicit "penalty" or "pseudocompressibility" methods of solving incompressible flows [11] [12] [13]. Complete energy-conserving models that yield the correct form of the compressible continuity, momentum, and energy equations have been developed by Alexander, Chen, and Sterling [14] and by McNamara and Alder [15]. We note that for any of the LB models, the transport coefficients depend on the time step and lattice spacing.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Lattice Boltzmann Methods

Sterling

Chen

1996

Journal of Computational Physics

383

232

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The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered. Although the equation is a derivative of lattice gas automata, it may be interpreted as a Lagrangian finite-difference method for the numerical simulation of the discrete-velocity Boltzmann equation that makes use of a BGK collision operator. As a result, it is not surprising that numerical instability of lattice Boltzmann methods have been frequently encountered by researchers. We present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constantdensity uniform mean flow. The linear stability depends on the following parameters: the distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wavenumber of the perturbations. This parameter space is too large to compute the complete stability characteristics. We report some stability results for a subset of the parameter space for a 7-velocity hexagonal lattice, a 9-velocity square lattice and a 15-velocity cubic lattice. Results common to all three lattices are 1) the BGK relaxation time τ must be greater than 1 2 corresponding to positive shear viscosity, 2) there exists a maximum stable mean velocity for fixed values of the other parameters and 3) as τ is increased from 1 2 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger τ .

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Section: Introductionmentioning

confidence: 99%

Stability Analysis of Lattice Boltzmann Methods

Sterling

Chen

1996

Journal of Computational Physics

383

232

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“…However, for the sake of completeness, we explain the equivalence. Considering the evolution of the discrete-velocity gas, the (model) Boltzmann equations 3,15]|a statement of the conservation of the number of particles with a particular discrete-velocity|are @n a @t + c a rn a = Q a (n; n); a = 0; : : :; 26; (8) where Q a is the nonlinear collision operator and the left hand side represents streaming of particles with velocity c a . The zeroth, rst, and second order velocity moments of (8) give respectively, noting that the moments of the collision terms on the right-hand side vanish owing to the mass, momentum, and energy conserving nature of each collision, where the overbar denotes averaging with respect to the discrete-velocities.…”

Section: The Equivalence Of the Model To The Euler Equationsmentioning

confidence: 99%

“…Such a discretization of the velocity space and de nition of the particle interactions (collisions or relaxation or more generally redistribution) also form the basis for the lattice gas and lattice Boltzmann techniques which have been developed over the last eight years 4,5,6 and references therein]. In spite of the inherently compressible nature of discrete-velocity gases, lattice gases and lattice-Boltzmann techniques, their applications in uid ow modeling have been restricted to the incompressible or very low Mach number regimes 4,5,6 and references therein, 7,8,9]. In this article, we present a discrete-velocity gas which for the rst time solves the compressible Euler equations without introducing any artifacts of the velocity discretization over a wide range of Mach numbers.…”

Section: Introductionmentioning

confidence: 99%

An Euler solver based on locally adaptive discrete velocities

Nadiga

1995

J Stat Phys

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A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature|both the origin of the discrete-velocity space and the magnitudes of the discrete-velocities are dependent on the local ow| and are used in a nite volume context. The numerical implementation of the model follows the near-equilibrium ow method of Nadiga and Pullin 1] and results in a scheme which is second order in space (in the smooth regions and between rst and second order at discontinuities) and second order in time.(The three-dimensional code is included.) For one choice of the scaling between the magnitude of the discrete-velocities and the local internal energy of the ow, the method reduces to a uxsplitting scheme based on characteristics. As a preliminary exercise, the result of the Sod shock-tube simulation is compared to the exact solution.

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“…In this paper we address the stability problem that arises when the temperature T is allowed to dynamically vary in LB [6,7]. This so-called thermal LB method is stable only for relatively large transport coefficients; i.e., viscosity and thermal conductivity.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, methods that stabilize thermal LB by modifying the collision operator are referred to as modified equation methods. An example of a modified equation method is increasing the number of discrete velocities beyond the requirement for correct hydrodynamics, which increases the computational expenses [6,[19][20][21][22][23][24]. Thermal LB has also been stabilized by reducing the velocity set below this requirement [14,25].…”

Section: Introductionmentioning

confidence: 99%

Stabilizing the thermal lattice Boltzmann method by spatial filtering

Gillissen

2016

Phys. Rev. E

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We propose to stabilize the thermal lattice Boltzmann method by filtering the second-and third-order moments of the collision operator. By means of the Chapman-Enskog expansion, we show that the additional numerical diffusivity diminishes in the low-wavnumber limit. To demonstrate the enhanced stability, we consider a threedimensional thermal lattice Boltzmann system involving 33 discrete velocities. Filtering extends the linear stability of this thermal lattice Boltzmann method to 10-fold smaller transport coefficients. We further demonstrate that the filtering does not compromise the accuracy of the hydrodynamics by comparing simulation results to reference solutions for a number of standardized test cases, including natural convection in two dimensions.

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Analysis of the lattice Boltzmann treatment of hydrodynamics

Cited by 153 publications

References 10 publications

Stability Analysis of Lattice Boltzmann Methods

Stability Analysis of Lattice Boltzmann Methods

An Euler solver based on locally adaptive discrete velocities

Stabilizing the thermal lattice Boltzmann method by spatial filtering

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