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@ARTICLE{bouzidi:01,
author = {Bouzidi, {M'hamed} and Firdaouss, Mouaouia and Lallemand, Pierre},
title = {Momentum transfer of a {B}oltzmann-lattice fluid with boundaries},
journal = {Phys. Fluids},
year = {2001},
volume = {13},
pages = {3452--3459},
doi = {10.1063/1.1399290},
category = {theory, boundary_condition},
comment = {The Bouzidi boundary condition is a common choice for curved or off-lattice boundaries. It is a variant of the [[models:bc|half-way bounce-back]] scheme, with extension from no-slip to a general Dirichlet boundary by addition of momentum, and an adaptation of the exact boundary location by use of interpolations.},
}
@ARTICLE{chen:91,
author = {Chen, Shiyi and Chen, Hudong and Martinez, Daniel and Matthaeus, William },
title = {Lattice {B}oltzmann model for simulation of magnetohydrodynamics},
journal = {Phys. Rev. Lett.},
year = {1991},
volume = {67},
pages = {3776--3779},
doi = {10.1103/PhysRevLett.67.3776},
category = {theory},
comment = {none},
}
@ARTICLE{chen:98,
author = {Chen, Shiyi and Doolen, Gary D.},
title = {Lattice {B}oltzmann Method for Fluid Flows},
journal = {Ann. Rev. Fluid Mech.},
year = {1998},
volume = {30},
pages = {329--364},
doi = {10.1146/annurev.fluid.30.1.329},
category = {theory, review},
comment = {A good review article on the BGK method, and on its various fields of application. The article summarizes important topics such as the implementation of boundary conditions, simulation of fluid turbulence, and multiphase/multicomponent flows.},
}
@ARTICLE{chopard:02,
author = {Chopard, Bastien and Dupuis, Alexandre and Masselot, Alexandre and Luthi, Pascal},
title = {Cellular Automata and Lattice {B}oltzmann techniques: an approach to model and simulate complex systems},
journal = {Adv. Compl. Sys.},
year = {2002},
volume = {5},
pages = {103--246},
doi = {10.1142/S0219525902000602},
category = {theory, review},
comment = {A review of the lattice Boltzmann method with numerous fields of application. Of particular interest is the thorough derivation of the BGK model as an asymptotic approximation of a Cellular Automaton. Also unique is the discussion on how to vary the speed of sound in the BGK model.},
}
@ARTICLE{clercx:05,
author = {Clercx, H. J. H and Bruneau, C.-H.},
title = {The normal and oblique collision of a dipole with a no-slip boundary},
journal = {Comp. Fluids},
year = {2006},
volume = {35},
pages = {245--279},
doi = {10.1016/j.compfluid.2004.11.009},
category = {non_lb},
comment = {none},
}
@ARTICLE{dellar:01,
author = {Dellar, Paul J.},
title = {Bulk and shear viscosities in lattice {B}oltzmann equations},
journal = {Phys. Rev. E},
year = {2001},
volume = {64},
pages = {031203},
doi = {10.1103/PhysRevE.64.031203},
category = {theory, model},
comment = {none},
}
@ARTICLE{dellar:03,
author = {Dellar, Paul J.},
title = {Incompressible limits of lattice {B}oltzmann equations using multiple relaxation times},
journal = {J. Comp. Phys.},
year = {2003},
volume = {190},
pages = {351--370},
doi = {10.1016/S0021-9991(03)00279-1},
category = {theory, analysis},
comment = {This article analyses Multiple-Relaxation-Time (MRT) models and shows that they can be deficient in a low Mach-number regime. Additionally to this, the article is also a great reference to various topics in lattice Boltzmann, thanks to its extensive introduction and discussions. It explains in a precise and easily understandable way what MRT models are, how lattice Boltzmann equations are derived from the continuum Boltzmann equation, how the accuracy of a simulation can be increased using a trick by P. Skordos, and other useful stuff.},
}
@ARTICLE{dhumieres:01,
author = {d'{Humi\`eres}, Dominique and Bouzidi, M'hamed and Lallemand, Pierre},
title = {Thirteen-velocity three-dimensional lattice {B}oltzmann model},
journal = {Phys. Rev. E},
year = {2001},
volume = {63},
pages = {066702},
doi = {10.1103/PhysRevE.63.066702},
category = {theory, model},
comment = {none},
}
@ARTICLE{dhumieres:02,
author = {d'{Humi\`eres}, Dominique and Ginzburg, Irina and Krafczyk, Manfred and Lallemand, Pierre and Luo, Li-Shi},
title = {Multiple-Relaxation-Time Lattice {B}oltzmann Models in Three Dimensions},
journal = {Phil. Trans. R. Soc. A},
year = {2002},
volume = {360},
pages = {437--451},
doi = {none},
category = {theory, model},
comment = {This article features an overview of the probably most commonly used Multiple-Relaxation-Time (MRT) model. In this model, the relaxation times of the ghost modes are given by ad-hoc numerical values. These values are derived from a linear stability analysis, and are mentioned explicitly in the article. The presented models are extensions of the "incompressible lattice Boltzmann" scheme. In stationary flows, they exhibit therefore a smaller compressibility error than fully compressible schemes (such as BGK). On the other hand, they have lost the ability of the BGK and related models to simulate weakly compressible flows. The article is implementation-oriented and delivers all required information to write a MRT code.},
}
@ARTICLE{geller:06,
author = {Geller, Sebastian and Krafczyk, Manfred and {T\"olke}, Jonas and Turek, Stefan and Hron, Jaroslav},
title = {Benchmark computations based on lattice-{B}oltzmann, finite element and finite volume methods for laminar flows},
journal = {Comp. Fluids},
year = {2006},
volume = {35},
pages = {888--897},
doi = {10.1016/j.compfluid.2005.08.009},
category = {application},
comment = {Is lattice Boltzmann more or less efficient than another method? This is an endless discussion, and the answer depends on what you are looking at. This paper provides you however with interesting comparisons that can help you decide which method to use for a given purpose.},
}
@ARTICLE{guo:02b,
author = {Guo, Zhaoli and Zheng, Chuguang and Shi, Baochang},
title = {Discrete lattice effects on the forcing term in the lattice {B}oltzmann method},
journal = {Phys. Rev. E},
year = {2002},
volume = {65},
pages = {046308},
doi = {10.1103/PhysRevE.65.046308},
category = {theory, body_force},
comment = {Many ways of implementing a body force in lattice Boltzmann seem intuitively right. A closer analysis shows however that only few of them lead to the expected asymptotic behavior with sufficient accuracy. This article analyzes different approaches and points out one method that is fully consistent with the hydrodynamic limit of the LB model.},
}
@ARTICLE{guo:02,
author = {Guo, Zhaoli and Zheng, Chuguang and Shi, Baochang},
title = {An extrapolation method for boundary conditions in lattice {B}oltzmann method},
journal = {Phys. Fluids},
year = {2002},
volume = {14},
pages = {2007--2010},
doi = {10.1063/1.1471914},
category = {theory, boundary_condition},
comment = {none},
}
@ARTICLE{guo:02c,
author = {Guo, Zhaoli and Shi, Baochang and Zheng, Chuguang},
title = {A coupled lattice {BGK} model for the {B}oussinesq equations},
journal = {Int. J. Num. Meth. Fluids},
year = {2002},
volume = {39},
pages = {325--342},
doi = {10.1002/fld.337},
category = {theory, thermal},
comment = {The Boltzmann equation describes the full statistical properties of a fluid, including temperature effects. But BGK and related models can only simulate isothermal fluids, because the lattice (D2Q9 or D3Q13-D3Q27) lacks sufficient symmetries to include thermal effects. This deficiency can be circumvented by using higher-order discretizations of Boltzmann equation, using a grid with higher connectivity. Yet another solution is to solve the temperature equation separately, as it is common in computational fluid dynamics, and to couple fluid and temperature appropriately. Such an approach is described in the paper by Guo, Shi, and Zheng. They simulate the fluid with a BGK model and the temperature field with a LB model for the advection-diffusion equation. A Boussinesq approximation is used to represent the action of temperature on the fluid by a linear buoyancy term.},
}
@ARTICLE{halliday:02,
author = {Halliday, I. and Hammond, L. A. and Care, C. M.},
title = {Enhanced closure scheme for lattice {B}oltzmann equation hydrodynamics},
journal = {J. Phys. A},
volume = {35},
year = {2002},
pages = {L157--L166},
doi = {10.1088/0305-4470/35/12/102},
category = {theory, boundary_condition},
comment = {none},
}
@ARTICLE{he:97,
author = {He, Xiaoyi and Luo, Li-Shi},
title = {Theory of the lattice {B}oltzmann method : From the {B}oltzmann equation to the lattice {B}oltzmann equation},
journal = {Phys. Rev. E},
year = {1997},
volume = {56},
pages = {6811--6818},
doi = {10.1103/PhysRevE.56.6811},
category = {theory, analysis},
comment = {This is one of the first papers describing the discretization of the continuous Boltzmann-BGK equation to recover the lattice Boltzmann scheme. The paper is mainly of theoretical interest since it does not address implementation issues. Please note that the developments in this paper only recover first-order time accuracy of the scheme, whereas it was later proved to be second-order accurate, as shown in [[literature:he_98|He et al. 98]] or in [[literature:dellar_03|Dellar 2003]].},
}
@ARTICLE{he:98,
author = {He, Xiaoyi and Shan, Xiaowen and Doolen, Gary D.},
title = {Discrete Boltzmann equation model for nonideal gases},
journal = {Phys. Rev. E},
year = {1998},
volume = {57},
pages = {R13--R16},
doi = {10.1103/PhysRevE.57.R13},
category = {theory, analysis},
comment = {This paper shows among other things how to get the lattice Boltzmann equation from the discrete Boltzmann equation. The trapezoid rule is used to integrate along the path of a fluid element in space and time, which shows that the numerical scheme is second-order accurate. It is therefore useful to read this paper together with [[literature:he_97|He et al. 1997]], because the path integration is only resolved with first-order accuracy in the latter. The whole "from discrete Boltzmann to lattice Boltzmann" business is also summarized nicely in [[literature:dellar_03|Dellar 2003]].},
}
@ARTICLE{inamuro:95,
author = {Inamuro, Takaji and Yoshina, Masato and Ogino, Fumimaru},
title = {A non-slip boundary condition for lattice {B}oltzmann simulations},
journal = {Phys. Fluids},
year = {1995},
volume = {7},
pages = {2928--2930},
doi = {10.1063/1.868766},
category = {theory, boundary_condition},
comment = {The Inamuro boundary condition can be used to implement velocity or pressure conditions on straight boundaries. This method is exceptionally accurate, especially in 2D flows, and regularly beats all other methods in benchmarks. It has however stability issues when the Reynolds number is increased, and 3D extensions of the model are not straightforward.},
}
@ARTICLE{junk:05,
author = {Junk, Michael and Klar, Axel and Luo, Li-Shi},
title = {Asymptotic analysis of the lattice {B}oltzmann equation},
journal = {J. Comput. Phys.},
volume = {210},
year = {2005},
pages = {676--704},
doi = {10.1016/j.jcp.2005.05.003 },
category = {theory, analysis},
comment = {none},
}
@ARTICLE{latt:06,
author = {Latt, Jonas and Chopard, Bastien},
title = {Lattice {B}oltzmann Method with regularized non-equilibrium distribution functions},
journal = {Math. Comp. Sim.},
year = {2006},
volume = {72},
pages = {165--168},
doi = {10.1063/1.868766},
category = {theory, model},
comment = {This article introduces the regularized LB method, one of the extensions of BGK which are more accurate and more stable for many problems. This and other models are summarized on the [[models:lbmodels|page on LB models]].},
}
@ARTICLE{latt:08,
author = {Latt, Jonas and Chopard, Bastien and Malaspinas, Orestis and Deville, Michel and Michler, Andreas},
title = {Straight velocity boundaries in the lattice {B}oltzmann method},
journal = {Phys. Rev. E},
year = {2008},
volume = {77},
pages = {056703},
doi = {10.1103/PhysRevE.77.056703},
category = {theory, boundary_condition},
comment = {This is a review paper for boundary conditions on straight walls in the "wet node" category (see [[models:bc|overview of boundary conditions]]). The following four boundary conditions are presented along with directions for their algorithmic implementation: the Inamuro boundary condition, the Zou/He boundary condition, the regularized boundary condition, and the non-local Skordos boundary condition. They are compared on an equal footing, using a common theoretical framework. Furthermore, their accuracy and stability is challenged numerically in various 2D and 3D benchmarks.},
}
@ARTICLE{mcnamara_88,
author = {McNamara, Guy R. and Zanetti, Gianluigi },
title = {Use of the {B}oltzmann Equation to Simulate Lattice-Gas Automata},
journal = {Phys. Rev. Lett.},
volume = {61},
pages = {2332--2335},
year = {1988},
doi = {10.1103/PhysRevLett.61.2332},
category = {theory, model},
comment = {This paper proposes a novel way to deal with the statistical noise of the Cellular Automata approach by replacing the boolean variables (which represent particles) by real numbers on each lattice site. The result is encouraging, as the computational cost is decreased at still acceptable accuracy. This is the first paper, to our knowledge, dealing with the lattice Boltzmann method.},
}
@ARTICLE{mcnamara:97,
author = {McNamara, Guy R. and Garcia, Alejandro L. and Alder, Berni J.},
title = {A hydrodynamically correct thermal lattice {B}oltzmann model},
journal = {J. Stat. Phys.},
volume = {87},
year = {1997},
pages = {1111--1121},
doi = {10.1007/BF02181274},
category = {theory, thermal},
comment = {none},
}
@ARTICLE{mei:06,
author = {Mei, Renwei and Luo, Li-Shi and Lallemand, Pierre and
d'{Humi\`eres}, Dominique},
title = {Consistent initial conditions for lattice {B}oltzmann
simulations},
journal = {Comp. Fluids},
year = {2006},
volume = {35},
pages = {855--862},
doi = {10.1016/j.compfluid.2005.08.008},
category = {theory, initial_condition},
comment = {An algorithm is introduced to generate an initial condition
for lattice Boltzmann simulations for which an initial
velocity field is prescribed. The basic idea is to run a few
initial lattice Boltzmann iterations, but to use the
prescribed velocity instead of the moment of particle
populations when computing the equilibrium distribution.},
}
@ARTICLE{qian:92,
author = {Qian, Y.H. and d'{Humi\`eres}, Dominique and Lallemand, Pierre},
title = {Lattice BGK Models for Navier-Stokes Equation},
journal = {Europhys. Lett.},
volume = {17},
year = {1992},
pages = {479--484},
doi = {10.1209/0295-5075/17/6/001},
category = {theory, model},
comment = {none},
}
@ARTICLE{qian:93,
author = {Qian, Y. H. and Orszag, S. A.},
title = {Lattice {BGK} Models for the {N}avier-{S}tokes Equation: Nonlinear Deviation in Compressible Regimes},
journal = {Europhys. Lett.},
volume = {21},
year = {1993},
pages = {255--259},
doi = {10.1209/0295-5075/21/3/001},
category = {theory, compressible},
comment = {none},
}
@ARTICLE{shan:06,
author = {Shan, Xiaowen and Yuan, Xue-Feng and Chen, Hudong},
title = {Kinetic theory representation of hydrodynamics: a way beyond the {N}avier-{S}tokes equation},
journal = {J. Fluid Mech.},
volume = {550},
year = {2006},
pages = {413--441},
doi = {10.1017/S0022112005008153},
category = {theory, analysis},
comment = {This paper proposes a systematic approach to deriving a discrete lattice Boltzmann scheme from the continuum Boltzmann equation. Inspired by the Grad-13 moment system, the Boltzmann distribution function is expanded on a Hermite basis in velocity space. Unlike Grad's approach, the paper focuses however not on the moment representation, but on a discrete set of kinetic variables, which are recovered by a reverse calculation from the set of moments. A Gauss-Hermite quadrature is used to integrate the hydrodynamic moments. It is shown that the discrete velocity set of the lattice Boltzmann scheme is identical with the quadrature points of this procedure. The paper extends previous results in He and Luo 97 and He et al. 98 to create a consistent theoretical framework for the lattice Boltzmann method. In particular, the D2Q17 and D3Q39 lattices are introduced, in which thermal effects or phenomena at higher Knudsen number can be modeled thanks to an enlarged neighborhood for inter-cell communication.},
}
@ARTICLE{shan:98,
author = {Shan, Xiaowen and He, Xiaoyi },
title = {Discretization of the Velocity Space in the Solution of the {B}oltzmann Equation},
journal = {Phys. Rev. Lett.},
volume = {80},
year = {1998},
pages = {65--68},
doi = {10.1103/PhysRevLett.80.65},
category = {theory, analysis},
comment = {none},
}
@ARTICLE{shan_chen:93,
author = {Shan, Xiaowen and Chen, Hudong},
title = {Lattice {B}oltzmann model for simulating flows with multiple phases and components},
journal = {Phys. Rev. E},
year = {1993},
volume = {47},
pages = {1815--1819},
doi = {10.1103/PhysRevE.47.1815},
category = {theory, multi_phase, multi_component},
comment = {This paper introduces the well-established Shan-Chen model for multicomponent and multiphase fluids. A local body-force is added to account for the intermolecular interaction between two fluid components or phases. The model is straightforward, and impressive results are obtained with fairly little effort. The Shan-Chen model is therefore widely used, despite potential numerical deficiencies in some regimes (it may for example be unstable for high density ratios between fluid components).},
}
@ARTICLE{skordos:93,
author = {Skordos, P. A.},
title = {Initial and boundary conditions for the lattice {B}oltzmann method},
journal = {Phys. Rev. E},
year = {1993},
volume = {48},
pages = {4823--4842},
doi = {10.1103/PhysRevE.48.4823},
category = {theory, boundary_condition, initial_condition},
comment = {This is probably the first article containing a systematic discussion on how to implement on-lattice boundary conditions in lattice Boltzmann using a finite difference scheme. The particle populations on the boundary are split into equilibrium and off-equilibrium part, and the results of the Chapman-Enskog expansion are used to associate off-equilbrium parts with velocity gradients. These gradient in their turn are evaluated by finite differences from the value of the velocity on neighboring nodes. It may take a while to get through this paper at a first reading, because the notation for the BGK model differs from the commonly used modern notation. However, the findings of the article remain valid and have been reused for different types of boundary conditions, including off-lattice boundaries.},
}
@ARTICLE{sun:00,
author = {Sun, Chenghai},
title = {Simulations of Compressible Flows with Strong Shocks by an Adaptive Lattice {B}oltzmann Model},
journal = {J. Comput. Phys.},
volume = {161},
year = {2000},
pages = {70--84},
doi = {10.1006/jcph.2000.6487},
category = {theory, compressible},
comment = {none},
}
@ARTICLE{teixeira:98,
author = {Teixeira, Christopher M.},
title = {Incorporating Turbulence Models into the Lattice-{B}oltzmann Method},
journal = {Int. J. Mod. Phys. C},
year = {1998},
volume = {9},
pages = {1159--1175},
doi = {10.1142/S0129183198001060},
category = {theory, turbulence},
comment = {This paper presents a numerical experiment of a turbulent flow. The fluid is simulated with lattice Boltzmann, and the two equations of a k-epsilon model are solved separately as a closure for the physics of unresolved subgrid scales. The results show that lattice Boltzmann works fine with the k-epsilon model and thus can be used for simulating high Reynolds-number flows.},
}
@ARTICLE{wang:06,
author = {Wang, J and Wang, M and Li, ZX},
title = {Lattice Poisson-Boltzmann Simulations of Electro-osmotic
Flows in Microchannels},
journal = {J. Colloid Interface Sci.},
year = {2006},
volume = {296},
pages = {729-736},
doi = {10.1016/j.jcis.2005.09.042},
category = {theory},
comment = {none},
}
@ARTICLE{zhao:93,
author = {Zhao, Xiao-Hua and Kwek, Keng-Huat and Li, Ji-Bin and Huang, Ke-Lei},
title = {Chaotic and resonant streamlines in the {ABC} flow},
journal = {SIAM J. Appl. Math.},
year = {1993},
volume = {53},
pages = {71--77},
doi = {10.1137/0153005},
category = {non_lb},
comment = {none},
}
@ARTICLE{zou_he:97,
author = {Zou, Qisu and He, Xiaoyi},
title = {On pressure and velocity boundary conditions for the lattice {B}oltzmann {BGK} model},
journal = {Phys. Fluids},
volume = {9},
year = {1997},
pages = {1591--1598},
doi = {10.1063/1.869307},
category = {theory, boundary_condition},
comment = {Like the [[literature:inamuro_95|Inamuro model]], the Zou/He model is very accurate, especially in 2D flows. In most benchmarks, it tends to be slightly less accurate, but also slightly more stable than the Inamuro approach. The 3D extension of the Zou/He model is quite straightforward, which is a great advantage of this approach.},
}