Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Serván-Camas, Borja; Tsai, Frank T.-C.

doi:10.1016/j.jcp.2008.09.005

Cited by 59 publications

(31 citation statements)

References 31 publications

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“…In common with other authors using similar Fourier techniques [16,17], and with our own earlier assumptions, we find that two lattice parameters are critical for stability. These are the time step which must be positive, and the relaxation parameter ω, which must be chosen for non-zero flow speed in the interval (−1, 1).…”

Section: Resultssupporting

confidence: 91%

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Packwood

Levesley

Gorban

2010

Lecture Notes in Computational Science and Engineering

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Summary. The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme.In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system.By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space.Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.

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Section: Resultssupporting

confidence: 91%

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Packwood

Levesley

Gorban

2010

Lecture Notes in Computational Science and Engineering

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“…[12] to model the nuclear envelope. However, if the membrane restricts the transport strongly, relaxation time at the membrane sites can be inconveniently small (with respect to the numerical stability [13]), or too large in the surrounding environment (with respect to the numerical error [11]). A possible way to circumvent this problem is to use a local grid refinement at the membrane and its immediate neighborhood.…”

Section: Lattice-boltzmann Modeling Of Diffusion Across a Membranementioning

confidence: 99%

Diffusion through thin membranes: Modeling across scales

et al. 2016

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From macroscopic to microscopic scales it is demonstrated that diffusion through membranes can be modeled using specific boundary conditions across them. The membranes are here considered thin in comparison to the overall size of the system. In a macroscopic scale the membrane is introduced as a transmission boundary condition, which enables an effective modeling of systems that involve multiple scales. In a mesoscopic scale, a numerical lattice-Boltzmann scheme with a partial-bounceback condition at the membrane is proposed and analyzed. It is shown that this mesoscopic approach provides a consistent approximation of the transmission boundary condition. Furthermore, analysis of the mesoscopic scheme gives rise to an expression for the permeability of a thin membrane as a function of a mesoscopic transmission parameter. In a microscopic model, the mean waiting time for a passage of a particle through the membrane is in accordance with this permeability. Numerical results computed with the mesoscopic scheme are then compared successfully with analytical solutions derived in a macroscopic scale, and the membrane model introduced here is used to simulate diffusive transport between the cell nucleus and cytoplasm through the nuclear envelope in a realistic cell model based on fluorescence microscopy data. By comparing the simulated fluorophore transport to the experimental one, we determine the permeability of the nuclear envelope of HeLa cells to enhanced yellow fluorescent protein.

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“…The most gentle solution gives the "positivity rule" [3,25,36,33]: to take the maximal value of α that guarantees f i + α( f * i − f i ) ≥ 0 for all i. In general, the Ehrenfest rule prescribes to send the most non-equilibrium sites to equilibrium and the positivity rule is applied for any LBM as a recommendation to substitute the non-positive vectors f by the closest non-negative on the interval of the straight line [f, f * ] that connects f to equilibrium.…”

Section: ¿º½º ä å û ø H ø óö ñmentioning

confidence: 99%

“…If f − f eq is too big, then the limiter should decrease its norm. For the first example of the realization of this pointwise filtering we use the kinetic idea of the positivity rule, the prescription is simple [3,25,36,33]: to substitute nonpositive F(f) by the closest nonnegative state that belongs to the straight line…”

Section: Entropic Filteringmentioning

confidence: 99%

Add-ons for Lattice Boltzmann Methods: Regularization, Filtering and Limiters

Brownlee¹,

Levesley²,

Packwood³

et al. 2013

Progress in Computational Physics Volume 3: Novel Trends in Lattice-Boltzmann Methods

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We describe how regularization of lattice Boltzmann methods can be achieved by modifying dissipation. Classes of techniques used to try to improve regularization of LBMs include flux limiters, enforcing the exact correct production of entropy and manipulating non-hydrodynamic modes of the system in relaxation. Each of these techniques corresponds to an additional modification of dissipation compared with the standard LBGK model. Using some standard 1D and 2D benchmarks including the shock tube and lid driven cavity, we explore the effectiveness of these classes of methods.

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Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Cited by 59 publications

References 31 publications

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Time Step Expansions and the Invariant Manifold Approach to Lattice Boltzmann Models

Diffusion through thin membranes: Modeling across scales

Add-ons for Lattice Boltzmann Methods: Regularization, Filtering and Limiters

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