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Journal articles on the topic 'Lattice Boltzmann scheme'

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Published: 16 September 2023

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1

Dubois, François, and Pierre Lallemand. "On Triangular Lattice Boltzmann Schemes for Scalar Problems." Communications in Computational Physics 13, no. 3 (March 2013): 649–70. http://dx.doi.org/10.4208/cicp.381011.270112s.

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AbstractWe propose to extend the d’Humieres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

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2

Venturi, Sara, Silvia Di Francesco, Martin Geier, and Piergiorgio Manciola. "Forcing for a Cascaded Lattice Boltzmann Shallow Water Model." Water 12, no. 2 (February 6, 2020): 439. http://dx.doi.org/10.3390/w12020439.

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This work compares three forcing schemes for a recently introduced cascaded lattice Boltzmann shallow water model: a basic scheme, a second-order scheme, and a centred scheme. Although the force is applied in the streaming step of the lattice Boltzmann model, the acceleration is also considered in the transformation to central moments. The model performance is tested for one and two dimensional benchmarks.

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3

Xu, Kun, and Li-Shi Luo. "Connection Between Lattice-Boltzmann Equation and Beam Scheme." International Journal of Modern Physics C 09, no. 08 (December 1998): 1177–87. http://dx.doi.org/10.1142/s0129183198001072.

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In this paper we analyze and compare the lattice-Boltzmann equation with the beam scheme in detail. We notice the similarity and differences between the lattice Boltzmann equation and the beam scheme. We show that the accuracy of the lattice-Boltzmann equation is indeed second order in space. We discuss the advantages and limitations of the lattice-Boltzmann equation and the beam scheme. Based on our analysis, we propose an improved multi-dimensional beam scheme.

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4

Gao, Shangwen, Chengbin Zhang, Yingjuan Zhang, Qiang Chen, Bo Li, and Suchen Wu. "Revisiting a class of modified pseudopotential lattice Boltzmann models for single-component multiphase flows." Physics of Fluids 34, no. 5 (May 2022): 057103. http://dx.doi.org/10.1063/5.0088246.

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Since its emergence, the pseudopotential lattice Boltzmann (LB) method has been regarded as a straightforward and practical approach for simulating single-component multiphase flows. However, its original form always results in a thermodynamic inconsistency, which, thus, impedes its further application. Several strategies for modifying the force term have been proposed to eliminate this limitation. In this study, four typical and widely used improved schemes—Li's single-relaxation-time (SRT) scheme [Li et al., “Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows,” Phys. Rev. E 86, 016709 (2012)] and multiple-relaxation-times (MRT) scheme [Li et al., “Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model,” Phys. Rev. E 87, 053301 (2013)], Kupershtokh's SRT scheme [Kupershtokh et al., “On equations of state in a lattice Boltzmann method,” Comput. Math. Appl. 58, 965 (2009)], and Huang's MRT scheme [Huang and Wu, “Third-order analysis of pseudopotential lattice Boltzmann model for multiphase flow,” J. Comput. Phys. 327, 121 (2016)]—are systematically analyzed and intuitively compared after an extension of the MRT framework. The theoretical and numerical results both indicate that the former three schemes are specific forms of the last one, which thus help further understand the improvements of these pseudopotential LB models for achieving thermodynamic consistency. In addition, we modified the calculation of the additional source term in the LB evolution equation. Numerical results for stationary and moving droplets confirm the higher accuracy.

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5

van der Sman, R. G. M., and M. H. Ernst. "Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices." Journal of Computational Physics 160, no. 2 (May 2000): 766–82. http://dx.doi.org/10.1006/jcph.2000.6491.

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6

Qiu, Ruofan, Rongqian Chen, and Yancheng You. "An implicit-explicit finite-difference lattice Boltzmann subgrid method on nonuniform meshes." International Journal of Modern Physics C 28, no. 04 (April 2017): 1750045. http://dx.doi.org/10.1142/s0129183117500450.

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In this paper, an implicit-explicit finite-difference lattice Boltzmann method with subgrid model on nonuniform meshes is proposed. The implicit-explicit Runge–Kutta scheme, which has good convergence rate, is used for the time discretization and a mixed difference scheme, which combines the upwind scheme with the central scheme, is adopted for the space discretization. Meanwhile, the standard Smagorinsky subgrid model is incorporated into the finite-difference lattice Boltzmann scheme. The effects of implicit-explicit Runge–Kutta scheme and nonuniform meshes of present lattice Boltzmann method are discussed through simulations of a two-dimensional lid-driven cavity flow on nonuniform meshes. Moreover, the comparison simulations of the present method and multiple relaxation time lattice Boltzmann subgrid method are conducted qualitatively and quantitatively.

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7

Wen, Mengke, Weidong Li, and Zhangyan Zhao. "A hybrid scheme coupling lattice Boltzmann method and finite-volume lattice Boltzmann method for steady incompressible flows." Physics of Fluids 34, no. 3 (March 2022): 037114. http://dx.doi.org/10.1063/5.0085370.

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We present a new hybrid method coupling the adaptive mesh refinement lattice Boltzmann method (AMRLBM) and the finite-volume lattice Boltzmann method (FVLBM) to improve both the simulation efficiency and adaptivity for steady incompressible flows with complex geometries. The present method makes use of the domain decomposition, in which the FVLBM sub-domain is applied to the region adjacent to the walls, and is coupled to the lattice Boltzmann method (LBM) sub-domain in the rest of the flow field to enhance the ability of the LBM to deal with irregular geometries without sacrificing the high efficiency and accuracy property of the LBM. In the LBM sub-domain, a cell-centered lattice structure-based AMRLBM is used and, in the FVLBM sub-domain, the gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme-based FVLBM is adopted to reduce the numerical dissipation and enhance the efficiency of FVLBM. Moreover, not like the conventional LBM and Navier–Stokes equation solver-based hybrid schemes, the present hybrid scheme combines two kinds of lattice Boltzmann equation solvers, that is, AMRLBM and FVLBM, which makes the present scheme much simpler and better consistency than the conventional hybrid schemes. To assess the accuracy and efficacy of the proposed method, various benchmark studies, including the Kovasznay flow, the lid-driven cavity flow with Reynolds number [Formula: see text], 400, and 1000, and the steady flow past a cylinder with [Formula: see text] and 40, are also conducted. The numerical results show that the present scheme can be an efficient and reliable method for steady incompressible flows.

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8

LALLEMAND, PIERRE, and LI-SHI LUO. "HYBRID FINITE-DIFFERENCE THERMAL LATTICE BOLTZMANN EQUATION." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 41–47. http://dx.doi.org/10.1142/s0217979203017060.

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We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.

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9

Wang, Liang, Zhaoli Guo, Baochang Shi, and Chuguang Zheng. "Evaluation of Three Lattice Boltzmann Models for Particulate Flows." Communications in Computational Physics 13, no. 4 (April 2013): 1151–72. http://dx.doi.org/10.4208/cicp.160911.200412a.

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AbstractA comparative study is conducted to evaluate three types of lattice Boltzmann equation (LBE) models for fluid flows with finite-sized particles, including the lattice Bhatnagar-Gross-Krook (BGK) model, the model proposed by Ladd [Ladd AJC, J. Fluid Mech., 271, 285-310 (1994); Ladd AJC, J. Fluid Mech., 271, 311-339 (1994)], and the multiple-relaxation-time (MRT) model. The sedimentation of a circular particle in a two-dimensional infinite channel under gravity is used as the first test problem. The numerical results of the three LBE schemes are compared with the theoretical results and existing data. It is found that all of the three LBE schemes yield reasonable results in general, although the BGK scheme and Ladd’s scheme give some deviations in some cases. Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes. Regarding the computational efficiency, it is found that the BGK scheme is the most superior one, while the other two schemes are nearly identical. We also observe that the MRT scheme can unequivocally reduce the viscosity dependence of the wall correction factor in the simulations, which reveals the superior robustness of the MRT scheme. The superiority of the MRT scheme over the other two schemes is also confirmed by the simulation of the sedimentation of an elliptical particle.

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10

Van Der Sman, R. G. M. "Lattice-Boltzmann Scheme for Natural Convection in Porous Media." International Journal of Modern Physics C 08, no. 04 (August 1997): 879–88. http://dx.doi.org/10.1142/s0129183197000758.

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A lattice-Boltzmann scheme for natural convection in porous media is developed and applied to the heat transfer problem of a 1000 kg potato packaging. The scheme has features new to the field of LB schemes. It is mapped on a orthorhombic lattice instead of the traditional cubic lattice. Furthermore the boundary conditions are formulated with a single paradigm based upon the particle fluxes. Our scheme is well able to reproduce (1) the analytical solutions of simple model problems and (2) the results from cooling down experiments with potato packagings.

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11

SOFONEA, VICTOR, and ROBERT F. SEKERKA. "DIFFUSIVITY OF TWO-COMPONENT ISOTHERMAL FINITE DIFFERENCE LATTICE BOLTZMANN MODELS." International Journal of Modern Physics C 16, no. 07 (July 2005): 1075–90. http://dx.doi.org/10.1142/s0129183105007741.

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Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

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12

TAKADA, NAOKI, AKIO TOMIYAMA, and SHIGEO HOSOKAWA. "LATTICE BOLTZMANN SIMULATION OF INTERFACIAL DEFORMATION." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 179–82. http://dx.doi.org/10.1142/s0217979203017308.

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This study describes the numerical simulations of two-phase interfacial deformations using the binary fluid (BF) model in the lattice Boltzmann method (LBM), where a macroscopic fluid involves mesoscopic particles repeating collisions and propagations and an interface is reproduced in a self-organizing way by repulsive interaction between different kinds of particles. Schemes for the BF model are proposed to simulate motions of immiscible two phases with different mass densities. For higher Reynolds number, the finite difference-based lattice Boltzmann scheme is applied to the kinetic equations of particles, which include convection terms to reduce the diffusivity of each phase volume. In addition, two parameters are introduced into the BF model to adjust surface tension and interfacial thickness independently. The numerical results of three-dimensional bubble motion under gravity and two-dimensional droplet deformation under shear stress indicate that the lattice-Boltzmann BF model with the proposed schemes would be applicable to simulating interfacial dynamics of immiscible two-phase fluids.

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13

WANG, Y., Y. L. HE, T. S. ZHAO, G. H. TANG, and W. Q. TAO. "IMPLICIT-EXPLICIT FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD FOR COMPRESSIBLE FLOWS." International Journal of Modern Physics C 18, no. 12 (December 2007): 1961–83. http://dx.doi.org/10.1142/s0129183107011868.

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We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge–Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant–Friedriche–Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.

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14

MILLER, W. "CRYSTAL GROWTH KINETICS AND FLUID FLOW." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 227–30. http://dx.doi.org/10.1142/s0217979203017394.

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A new type of a lattice phase-field model is developed and coupled with the lattice Boltzmann method to compute the soldification influenced by convection. Two methods of treating the solid-fluid interaction within the lattice Boltzmann scheme are tested.

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15

ZHOU, JIAN GUO. "LATTICE BOLTZMANN SIMULATIONS OF DISCONTINUOUS FLOWS." International Journal of Modern Physics C 18, no. 01 (January 2007): 1–14. http://dx.doi.org/10.1142/s0129183107010280.

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The lattice Boltzmann model for the shallow water equations (LABSWE) is applied to the simulation of certain discontinuous flows. Curved boundaries are treated efficiently, using either the elastic-collision scheme for slip and semi-slip boundary conditions or the bounce-back scheme for no-slip conditions. The force term is accurately determined by means of the centred scheme. Simulations are presented of a small pulse-like perturbation of the still water surface, a dam break, and a surge wave interaction with a circular cylinder. The results agree well with predictions from alternative high-resolution Riemann solver based methods, demonstrating the capability of LABSWE to predict shallow water flows containing discontinuities.

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16

Succi, S., and P. Vergari. "A Lattice Boltzmann Scheme for Semiconductor Dynamics." VLSI Design 6, no. 1-4 (January 1, 1998): 137–40. http://dx.doi.org/10.1155/1998/54940.

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17

Ma, Huifang, Bin Wu, Ying Wang, Hao Ren, Wanshun Jiang, Mingming Tang, and Wenyue Guo. "A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium." Electronics 11, no. 6 (March 10, 2022): 882. http://dx.doi.org/10.3390/electronics11060882.

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A one-dimensional plasma medium is playing a crucial role in modern sensing device design, which can benefit significantly from numerical electromagnetic wave simulation. In this study, we introduce a novel lattice Boltzmann scheme with a single extended force term for electromagnetic wave propagation in a one-dimensional plasma medium. This method is developed by reconstructing the solution to the macroscopic Maxwell’s equations recovered from the lattice Boltzmann equation. The final formulation of the lattice Boltzmann scheme involves only the equilibrium and one non-equilibrium force term. Among them, the former is calculated from the macroscopic electromagnetic variables, and the latter is evaluated from the dispersive effect. Thus, the proposed lattice Boltzmann scheme directly tracks the evolution of macroscopic electromagnetic variables, which yields lower memory costs and facilitates the implementation of physical boundary conditions. Detailed conduction is carried out based on the Chapman–Enskog expansion technique to prove the mathematical consistency between the proposed lattice Boltzmann scheme and Maxwell’s equations. Based on the proposed method, we present electromagnetic pulse propagating behaviors in nondispersive media and the response of a one-dimensional plasma slab to incident electromagnetic waves that span regions above and below the plasma frequency ωp, and further investigate the optical properties of a one-dimensional plasma photonic crystal with periodic thin layers of plasma with different layer thicknesses to verify the stability, accuracy, and flexibility of the proposed method.

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18

XUAN, Yimin. "Application of lattice Boltzmann scheme to nanofluids." Science in China Series E 47, no. 2 (2004): 129. http://dx.doi.org/10.1360/03ye0163.

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19

Dubois, François, Pierre Lallemand, and Mahdi Tekitek. "On a superconvergent lattice Boltzmann boundary scheme." Computers & Mathematics with Applications 59, no. 7 (April 2010): 2141–49. http://dx.doi.org/10.1016/j.camwa.2009.08.055.

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20

Succi, S., M. Vergassola, and R. Benzi. "Lattice Boltzmann scheme for two-dimensional magnetohydrodynamics." Physical Review A 43, no. 8 (April 1, 1991): 4521–24. http://dx.doi.org/10.1103/physreva.43.4521.

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21

Krivovichev, Gerasim Vladimirovich. "On the stability of lattice boltzmann equations for one-dimensional diffusion equation." International Journal of Modeling, Simulation, and Scientific Computing 08, no. 01 (January 10, 2017): 1750013. http://dx.doi.org/10.1142/s1793962317500131.

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Stability analysis of lattice Boltzmann equations (LBEs) on initial conditions for one-dimensional diffusion is performed. Stability of the solution of the Cauchy problem for the system of linear Bhatnaghar–Gross–Krook kinetic equations is demonstrated for the cases of D1Q2 and D1Q3 lattices. Stability of the scheme for D1Q2 lattice is analytically analyzed by the method of differential approximation. Stability of parametrical scheme is numerically investigated by von Neumann method in parameter space. As a result of numerical analysis, the correction of the hypothesis on transfer of stability conditions of the scheme for macroequation to the system of LBEs is demonstrated.

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22

SUGA, SHINSUKE. "STABILITY AND ACCURACY OF LATTICE BOLTZMANN SCHEMES FOR ANISOTROPIC ADVECTION-DIFFUSION EQUATIONS." International Journal of Modern Physics C 20, no. 04 (April 2009): 633–50. http://dx.doi.org/10.1142/s0129183109013856.

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The stability of the numerical schemes for anisotropic advection-diffusion equations derived from the lattice Boltzmann equation with the D2Q4 particle velocity model is analyzed through eigenvalue analysis of the amplification matrices of the scheme. Accuracy of the schemes is investigated by solving benchmark problems, and the LBM scheme is compared with traditional implicit schemes. Numerical experiments demonstrate that the LBM scheme produces stable numerical solutions close to the analytical solutions when the values of the relaxation parameters in x and y directions are greater than 1.9 and the Courant numbers satisfy the stability condition. Furthermore, the numerical solutions produced by the LBM scheme are more accurate than those of the Crank–Nicolson finite difference scheme for the case where the Courant numbers are set to be values close to the upper bound of the stability region of the scheme.

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23

NOR AZWADI, C. S., and T. TANAHASHI. "SIMPLIFIED THERMAL LATTICE BOLTZMANN IN INCOMPRESSIBLE LIMIT." International Journal of Modern Physics B 20, no. 17 (July 10, 2006): 2437–49. http://dx.doi.org/10.1142/s0217979206034789.

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In this paper, an incompressible thermohydrodynamics for the lattice Boltzmann scheme is developed. The basic idea is to solve the velocity field and the temperature field using two different distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmann equation is discussed in detail. By using the same procedure as in the derivation of the discretised density distribution function, we found that a new lattice of four-velocity model for internal energy density distribution function can be developed where the viscous and compressive heating effects are negligible. This model is validated by the numerical simulation of the porous plate couette flow problem where the analytical solution exists and the natural convection flows in a square cavity.

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24

ZHOU, J. G. "AN ELASTIC-COLLISION SCHEME FOR LATTICE BOLTZMANN METHODS." International Journal of Modern Physics C 12, no. 03 (March 2001): 387–401. http://dx.doi.org/10.1142/s0129183101001833.

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An elastic-collision scheme is developed to achieve slip and semi-slip boundary conditions for lattice Boltzmann methods. Like the bounce-back scheme, the proposed scheme is efficient, robust and generally suitable for flows in arbitrary complex geometries. It involves an equivalent level of computation effort to the bounce-back scheme. The new scheme is verified by predicting wind-driven circulating flows in a dish-shaped basin and a flow in a strongly bent channel, showing good agreement with analytical solutions and experimental data. The capability of the scheme for simulating flows through multiple bodies has also been demonstrated.

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25

Zhang, Raoyang, Chenghai Sun, Yanbing Li, Rajani Satti, Richard Shock, James Hoch, and Hudong Chen. "Lattice Boltzmann Approach for Local Reference Frames." Communications in Computational Physics 9, no. 5 (May 2011): 1193–205. http://dx.doi.org/10.4208/cicp.021109.111110s.

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AbstractIn this paper we present a generalized lattice Boltzmann based approach for sliding-mesh local reference frame. This scheme exactly conserves hydrodynamic fluxes across local reference frame interface. The accuracy and robustness of our scheme are demonstrated by benchmark validations.

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26

BERNASCHI, MASSIMO, and SAURO SUCCI. "ACCELERATED LATTICE BOLTZMANN SCHEME FOR STEADY-STATE FLOWS." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 1–7. http://dx.doi.org/10.1142/s021797920301700x.

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Delouei, A. Amiri, M. Nazari, M. H. Kayhani, and S. Succi. "Immersed Boundary – Thermal Lattice Boltzmann Methods for Non-Newtonian Flows Over a Heated Cylinder: A Comparative Study." Communications in Computational Physics 18, no. 2 (July 30, 2015): 489–515. http://dx.doi.org/10.4208/cicp.060414.220115a.

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AbstractIn this study, we compare different diffuse and sharp interface schemes of direct-forcing immersed boundary — thermal lattice Boltzmann method (IB-TLBM) for non-Newtonian flow over a heated circular cylinder. Both effects of the discrete lattice and the body force on the momentum and energy equations are considered, by applying the split-forcing Lattice Boltzmann equations. A new technique based on predetermined parameters of direct forcing IB-TLBM is presented for computing the Nusselt number. The study covers both steady and unsteady regimes (20<Re<80) in the power-law index range of 0.6<n<1.4, encompassing both shear-thinning and shear-thickening non-Newtonian fluids. The numerical scheme, hydrodynamic approach and thermal parameters of different interface schemes are compared in both steady and unsteady cases. It is found that the sharp interface scheme is a suitable and possibly competitive method for thermal-IBM in terms of accuracy and computational cost.

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28

ZHENG, H. W., and C. SHU. "EVALUATION OF THE PERFORMANCE OF THE HYBRID LATTICE BOLTZMANN BASED NUMERICAL FLUX." International Journal of Modern Physics: Conference Series 42 (January 2016): 1660152. http://dx.doi.org/10.1142/s2010194516601526.

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It is well known that the numerical scheme is a key factor to the stability and accuracy of a Navier-Stokes solver. Recently, a new hybrid lattice Boltzmann numerical flux (HLBFS) is developed by Shu's group. It combines two different LBFS schemes by a switch function. It solves the Boltzmann equation instead of the Euler equation. In this article, the main object is to evaluate the ability of this HLBFS scheme by our in-house cell centered hybrid mesh based Navier-Stokes code. Its performance is examined by several widely-used bench-mark test cases. The comparisons on results between calculation and experiment are conducted. They show that the scheme can capture the shock wave as well as the resolving of boundary layer.

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Che Sidik, Nor Azwadi, and Aman Ali Khan. "Simulation of Flow over a Cavity Using Multi-Relaxation Time Thermal Lattice Boltzmann Method." Applied Mechanics and Materials 554 (June 2014): 296–300. http://dx.doi.org/10.4028/www.scientific.net/amm.554.296.

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This article provides numerically study of the multi-relaxation time thermal lattice Boltzmann method (LBM) for compute the flow and isotherm characteristics in the bottom heated cavity located o n a floor of horizontal channel . A double-distribution function (DFF) was coupled with MRT thermal LBM to study the effects of various grashof number (Gr), Reynolds number (Re) and Aspect Ratio (AR) on the flow and isotherm characteristic. The results we re compared with the conventional single-relaxation time lattice Boltzmann scheme and benchmark solution for such flow configuration. The results of the numer ical simulation indicate that multi-relaxation time thermal lattice Boltzmann scheme demonstrated good agreement, which supports its validity in computing fluid flow problem.

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Li, Qiaojie, Zhoushun Zheng, Shuang Wang, and Jiankang Liu. "A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method." Journal of Applied Mathematics 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/925920.

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An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmann method. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have been compared with the exact solutions reported in previous studies. TheL2, L∞and Root-Mean-Square (RMS) errors in the solutions show that the scheme is accurate and effective.

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31

Yahia, Eman, William Schupbach, and Kannan N. Premnath. "Three-Dimensional Central Moment Lattice Boltzmann Method on a Cuboid Lattice for Anisotropic and Inhomogeneous Flows." Fluids 6, no. 9 (September 10, 2021): 326. http://dx.doi.org/10.3390/fluids6090326.

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Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.

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32

Haussmann, Marc, Stephan Simonis, Hermann Nirschl, and Mathias J. Krause. "Direct numerical simulation of decaying homogeneous isotropic turbulence — numerical experiments on stability, consistency and accuracy of distinct lattice Boltzmann methods." International Journal of Modern Physics C 30, no. 09 (September 2019): 1950074. http://dx.doi.org/10.1142/s0129183119500748.

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Stability, consistency and accuracy of various lattice Boltzmann schemes are investigated by means of numerical experiments on decaying homogeneous isotropic turbulence (DHIT). Therefore, the Bhatnagar–Gross–Krook (BGK), the entropic lattice Boltzmann (ELB), the two-relaxation-time (TRT), the regularized lattice Boltzann (RLB) and the multiple-relaxation-time (MRT) collision schemes are applied to the three-dimensional Taylor–Green vortex, which represents a benchmark case for DHIT. The obtained turbulent kinetic energy, the energy dissipation rate and the energy spectrum are compared to reference data. Acoustic and diffusive scaling is taken into account to determine the impact of the lattice Mach number. Furthermore, three different Reynolds numbers [Formula: see text], [Formula: see text] and [Formula: see text] are considered. BGK shows instabilities, when the mesh is highly underresolved. The diverging simulations for MRT are ascribed to a strong lattice Mach number dependency. Despite the fact that the ELB modifies the bulk viscosity, it does not mimic a turbulence model. Therefore, no significant increase of stability in comparison to BGK is observed. The TRT “magic parameter” for DHIT at moderate Reynolds numbers is estimated with respect to the energy contribution. Stability and accuracy of the TRT scheme is found to be similar to BGK. For small lattice Mach numbers, the RLB scheme exhibits lowered energy contribution in the dissipation range compared to an analytical model spectrum. Overall, to enhance stability and accuracy, the lattice Mach number should be chosen with respect to the applied collision scheme.

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33

PAN, X. F., AIGUO XU, GUANGCAI ZHANG, and SONG JIANG. "LATTICE BOLTZMANN APPROACH TO HIGH-SPEED COMPRESSIBLE FLOWS." International Journal of Modern Physics C 18, no. 11 (November 2007): 1747–64. http://dx.doi.org/10.1142/s0129183107011716.

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We present an improved lattice Boltzmann model for high-speed compressible flows. The model is composed of a discrete-velocity model by Kataoka and Tsutahara15 and an appropriate finite-difference scheme combined with an additional dissipation term. With the dissipation term parameters in the model can be flexibly chosen so that the von Neumann stability condition is satisfied. The influence of the various model parameters on the numerical stability is analyzed and some reference values of parameter are suggested. The new scheme works for both subsonic and supersonic flows with a Mach number up to 30 (or higher), which is validated by well-known benchmark tests. Simulations on Riemann problems with very high ratios (1000:1) of pressure and density also show good accuracy and stability. Successful recovering of regular and double Mach shock reflections shows the potential application of the lattice Boltzmann model to fluid systems where non-equilibrium processes are intrinsic. The new scheme for stability can be easily extended to other lattice Boltzmann models.

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Dubois, François. "Third order equivalent equation of lattice Boltzmann scheme." Discrete and Continuous Dynamical Systems 23, no. 1/2 (September 2008): 221–48. http://dx.doi.org/10.3934/dcds.2009.23.221.

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Krivovichev, G. V. "On the finite-element-based lattice Boltzmann scheme." Applied Mathematical Sciences 8 (2014): 1605–20. http://dx.doi.org/10.12988/ams.2014.4138.

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Halliday, I., L. A. Hammond, and C. M. Care. "Enhanced closure scheme for lattice Boltzmann equation hydrodynamics." Journal of Physics A: Mathematical and General 35, no. 12 (March 15, 2002): L157—L166. http://dx.doi.org/10.1088/0305-4470/35/12/102.

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SETA, Takeshi, Koji KONO, Daniel MARTINEZ, and Shiyi CHEN. "Lattice Boltzmann Scheme for Simulating Two-Phase Flows." JSME International Journal Series B 43, no. 2 (2000): 305–13. http://dx.doi.org/10.1299/jsmeb.43.305.

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Ho, Jeng-Rong, Chun-Pao Kuo, Wen-Shu Jiaung, and Cherng-Jyh Twu. "LATTICE BOLTZMANN SCHEME FOR HYPERBOLIC HEAT CONDUCTION EQUATION." Numerical Heat Transfer, Part B: Fundamentals 41, no. 6 (June 2002): 591–607. http://dx.doi.org/10.1080/10407790190053798.

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SETA, Takeshi, Koji KONO, Daniel MARTINEZ, and Shiyi CHEN. "Lattice Boltzmann Scheme for Simulating Two-Phase Flows." Transactions of the Japan Society of Mechanical Engineers Series B 65, no. 634 (1999): 1955–63. http://dx.doi.org/10.1299/kikaib.65.1955.

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Tian, Zhi-Wei, Chun Zou, Hong-Juan Liu, Zhao-Li Guo, Zhao-Hui Liu, and Chu-Guang Zheng. "Lattice Boltzmann scheme for simulating thermal micro-flow." Physica A: Statistical Mechanics and its Applications 385, no. 1 (November 2007): 59–68. http://dx.doi.org/10.1016/j.physa.2007.01.021.

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41

Alvarez-Ramírez, José, Francisco J. Valdés-Parada, and J. Alberto Ochoa-Tapia. "A lattice-Boltzmann scheme for Cattaneo’s diffusion equation." Physica A: Statistical Mechanics and its Applications 387, no. 7 (March 2008): 1475–84. http://dx.doi.org/10.1016/j.physa.2007.10.051.

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Chen, Sheng, Zhaohui Liu, Zhiwei Tian, Baochang Shi, and Chuguang Zheng. "A simple lattice Boltzmann scheme for combustion simulation." Computers & Mathematics with Applications 55, no. 7 (April 2008): 1424–32. http://dx.doi.org/10.1016/j.camwa.2007.08.020.

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Stiebler, Maik, Jonas Tölke, and Manfred Krafczyk. "Advection–diffusion lattice Boltzmann scheme for hierarchical grids." Computers & Mathematics with Applications 55, no. 7 (April 2008): 1576–84. http://dx.doi.org/10.1016/j.camwa.2007.08.024.

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44

Xie, Wenjun. "An axisymmetric multiple-relaxation-time lattice Boltzmann scheme." Journal of Computational Physics 281 (January 2015): 55–66. http://dx.doi.org/10.1016/j.jcp.2014.10.019.

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45

Liu, Ning Ning. "The Numerical Solution of Richards Equation Using the Lattice Boltzmann Method." Applied Mechanics and Materials 188 (June 2012): 90–95. http://dx.doi.org/10.4028/www.scientific.net/amm.188.90.

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The Richards equation is applied to describe the unsaturated soil moisture movement. The Lattice Boltzmann method is developed to solve this partial differential equation. The accuracy and efficiency of the Lattice Boltzmann method in modeling unsaturated soil moisture movement are compared to the Philip series method as well as Crank-Nicolson finite difference scheme. The results reveal that all three methods provide solutions of comparable accuracy. The computation efficiency, accuracy and simplicity of the Lattice Boltzmann method indicate that it has the capacity to model unsaturated soil moisture movement.

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ZHOU, JIAN GUO. "MRT RECTANGULAR LATTICE BOLTZMANN METHOD." International Journal of Modern Physics C 23, no. 05 (May 2012): 1250040. http://dx.doi.org/10.1142/s0129183112500404.

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A multiple-relaxation-time (MRT) collision operator is introduced into the author's rectangular lattice Boltzmann method for simulating fluid flows. The model retains both the advantages and the standard procedure of using a constant transformation matrix in the conventional MRT scheme on a square lattice, leading to easy implementation in the algorithm. This allows flow problems characterized by dominant feature in one direction to be solved more efficiently. Two numerical tests have been carried out and shown that the proposed model is able to capture complex flow characteristics and generate an accurate solution if an appropriate lattice ratio is used. The model is found to be more stable compared to the original rectangular lattice Boltzmann method using the single relaxation time.

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47

Zarghami, A., M. J. Maghrebi, J. Ghasemi, and S. Ubertini. "Lattice Boltzmann Finite Volume Formulation with Improved Stability." Communications in Computational Physics 12, no. 1 (July 2012): 42–64. http://dx.doi.org/10.4208/cicp.151210.140711a.

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AbstractThe most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especially when there is a need for high resolutions near the body or the walls. Among the recent advances in lattice Boltzmann research to handle complex geometries, a particularly remarkable option is represented by changing the solution procedure from the original “stream and collide” to a finite volume technique. However, most of the presented schemes have stability problems. This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme. To enhance stability, upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice. The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers: boundary layer, mixing layer and plane jet flows. The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data. Results in self-similar coordinates are also investigated and show that the time-averaged statistics for velocity and vorticity express self-similarity at low Reynolds numbers. Furthermore, the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent. The agreement of the numerical results with previous results is satisfactory.

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Mendl, Christian B. "Matrix-valued quantum lattice Boltzmann method." International Journal of Modern Physics C 26, no. 10 (June 24, 2015): 1550113. http://dx.doi.org/10.1142/s0129183115501132.

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We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

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JI, C. Z., C. SHU, and N. ZHAO. "A LATTICE BOLTZMANN METHOD-BASED FLUX SOLVER AND ITS APPLICATION TO SOLVE SHOCK TUBE PROBLEM." Modern Physics Letters B 23, no. 03 (January 30, 2009): 313–16. http://dx.doi.org/10.1142/s021798490901828x.

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This paper presents an approach, which combines the conventional finite volume method (FVM) with the lattice Boltzmann Method (LBM), to simulate compressible flows. Similar to the Godunov scheme, in the present approach, LBM is used to evaluate the flux at the interface for local Riemann problem when solving Euler/Navier-Stokes (N-S) equations by FVM. Two kinds of popular compressible Lattice Boltzmann models are applied in the new scheme, and some numerical experiments are performed to validate the proposed approach. From the sharper shock profile and higher computational efficiency, numerical results demonstrate that the proposed scheme is superior to the conventional Godunov scheme. It is expected that the proposed scheme has a potential to become an efficient flux solver in solving compressible Euler/N-S equations.

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LAMURA, ANTONIO, and SAURO SUCCI. "A LATTICE BOLTZMANN FOR DISORDERED FLUIDS." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 145–48. http://dx.doi.org/10.1142/s0217979203017230.

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A variant of the lattice Boltzmann scheme is presented as a mesoscopic model of glassy behavior. A hierarchical density-dependent interaction potential is introduced, which allows the coexistence and competition of multiple density minima. We find that this competition allows to model geometrical frustration which produces disordered patterns with sharp density contrasts and no phase-separation. A way of modeling mechanical arrest of real glasses is also proposed and discussed.

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