Coarse-grained numerical bifurcation analysis of lattice Boltzmann models

Leemput, Pieter Van; Lust, Kurt; Kevrekidis, I. G.

doi:10.1016/j.physd.2005.06.033

Cited by 22 publications

(16 citation statements)

References 26 publications

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“…To this end, it is possible to perform a constrained simulation before initialization to create such mature initial conditions [9,15]. If this is not done, the resulting evolution may be far from what is expected, see [21] for an illustration in the case of a lattice-Boltzmann model.…”

Section: Discussionmentioning

confidence: 99%

Patch dynamics with buffers for homogenization problems

Samaey

Kevrekidis

Roose

2006

Journal of Computational Physics

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An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an "equation-free" framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the space-time domain. To reduce the effect of the artificially introduced box boundaries, we use buffer regions to "shield" the boundary artefacts from the interior of the domain for short time intervals. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity, and propose a simple heuristic to determine a sufficient buffer size. The algorithm performance is illustrated through a set of numerical examples, which include a non-linear reaction-diffusion equation and the Kuramoto-Sivashinsky equation.

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Section: Discussionmentioning

confidence: 99%

Patch dynamics with buffers for homogenization problems

Samaey

Kevrekidis

Roose

2006

Journal of Computational Physics

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“…1 (left)). Although this attraction is fast compared to the dominant time scales governing the behavior of the distribution functions afterwards, this may lead to a substantial and persistent error [32], as can be seen in Fig. 1 (bottom-right).…”

Section: A Lbm For One-dimensional Reaction-diffusion Systemsmentioning

confidence: 99%

“…This puts the η lattice points at the midpoints of the lattice intervals. As in [32], we use a LBM time step t = 0.001, unless explicitly stated otherwise. The diffusion coefficients in (4) are D ac = 1 and D in = 4, and the corresponding values of ω s are ω ac ≈ 1.54 and ω in ≈ 0.91.…”

Section: A Lbm For the Fitzhugh-nagumo Systemmentioning

confidence: 99%

Acceleration of lattice Boltzmann models through state extrapolation: a reaction–diffusion example

Vandekerckhove

Leemput

Roose

2008

Applied Numerical Mathematics

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“…An alternative and preferrable strategy (see e.g. [25]) when t is not required to be small is to fix k and vary t so as to enable T to be approximated more exactly. Eigenvalues of the Jacobian matrix G u (u * ) are approximations to the Floquet multipliers of the Monodromy matrix / u (u * , T ).…”

Section: A3 Time-periodic Solutionsmentioning

confidence: 99%

Adapting simulation codes to compute unstable solutions and bifurcation behaviour

Grande¹,

Tavener²,

Thomas³

2007

Numerical Methods in Fluids

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SUMMARYSimulation codes for solving large systems of ordinary differential equations suffer from the disadvantage that bifurcation-theoretic results about the underlying dynamical system cannot be obtained from them easily, if at all. Bifurcation behaviour typically can be inferred only after significant computational effort, and even then the exact location and nature of the bifurcation cannot always be determined definitively. By incorporating relatively minor changes to an existing simulation code for the Taylor-Couette problem, specifically, by implementing the Newton-Picard method, we have developed a computational structure that enables us to overcome some of the inherent limitations of the simulation code and begin to perform bifurcation-theoretic tasks. While a complete bifurcation picture was not developed, three distinct solution branches of the Taylor-Couette problem were analysed. These branches exhibit a wide variety of behaviours, including Hopf bifurcation points, symmetry-breaking bifurcation points, turning points and bifurcation to motion on a torus. Unstable equilibrium and time-periodic solutions were also computed.

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Coarse-grained numerical bifurcation analysis of lattice Boltzmann models

Cited by 22 publications

References 26 publications

Patch dynamics with buffers for homogenization problems

Patch dynamics with buffers for homogenization problems

Acceleration of lattice Boltzmann models through state extrapolation: a reaction–diffusion example

Adapting simulation codes to compute unstable solutions and bifurcation behaviour

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