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Journal articles on the topic 'Discrete Boltzmann equation'

Author: Grafiati

Published: 14 March 2023

Last updated: 19 July 2025

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1

Bobylev, Aleksandr Vasil'evich. "Boltzmann-type kinetic equations and discrete models." Russian Mathematical Surveys 79, no. 3 (2024): 459–513. http://dx.doi.org/10.4213/rm10161e.

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The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this end we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

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Bobylev, Aleksandr Vasil'evich. "Boltzmann-type kinetic equations and discrete models." Uspekhi Matematicheskikh Nauk 79, no. 3(477) (2024): 93–148. http://dx.doi.org/10.4213/rm10161.

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The known non-linear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim-Uehling-Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the kinetic equations mentioned above correspond to different forms of the function (polynomial) $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas which are similar to those used for the construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similar to the Boltzmann $H$-function. The existence and uniqueness theorem for global in time solution of the Cauchy problem for these models is proved. Moreover, it is proved that the solution converges to the equilibrium solution when time goes to infinity. The properties of the equilibrium solution and the connection with solutions of the wave kinetic equation are discussed. The problem of the approximation of the Boltzmann-type equation by its discrete models is also discussed. The paper contains a concise introduction to the Boltzmann equation and its main properties. In principle, it allows one to read the paper without any preliminary knowledge in kinetic theory. Bibliography: 61 titles.

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3

Bobylev, A. V. "On discrete models of Boltzmann-type kinetic equations." Contemporary Mathematics. Fundamental Directions 70, no. 1 (2024): 15–24. http://dx.doi.org/10.22363/2413-3639-2024-70-1-15-24.

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The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables \(F(x,y; v,w)\). The function \(F\) is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function \(F\). Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann \(H\)-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

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4

Simonis, Stephan, Martin Frank, and Mathias J. Krause. "On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2175 (2020): 20190400. http://dx.doi.org/10.1098/rsta.2019.0400.

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The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods to approximate different types of partial differential equations. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

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5

Bobylev, Alexander Vasilievich, and Sergei Borisovitch Kuksin. "Boltzmann equation and wave kinetic equations." Keldysh Institute Preprints, no. 31 (2023): 1–20. http://dx.doi.org/10.20948/prepr-2023-31.

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The well-known nonlinear kinetic equations (in particular, the wave kinetic equation and the quantum Nordheim – Uehling – Uhlenbeck equations) are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the generalized kinetic equation that depends on a function of four real variables F(x1; x2; x3; x4). The function F is assumed to satisfy certain commutation relations. The general properties of this equation are studied. It is shown that the above mentioned kinetic equations correspond to different forms of the function (polynomial) F. Then the problem of discretization of the generalized kinetic equation is considered on the basis of ideas which are similar to those used for construction of discrete models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models possses a monotone functional similar to Boltzmann H-function. The behaviour of solutions of the simplest Broadwell model for the wave kinetic equation is discussed in detail.

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6

QU, KUN, CHANG SHU, and JINSHENG CAI. "DEVELOPING LBM-BASED FLUX SOLVER AND ITS APPLICATIONS IN MULTI-DIMENSION SIMULATIONS." International Journal of Modern Physics: Conference Series 19 (January 2012): 90–99. http://dx.doi.org/10.1142/s2010194512008628.

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In this paper, a new flux solver was developed based on a lattice Boltzmann model. Different from solving discrete velocity Boltzmann equation and lattice Boltzmann equation, Euler/Navier-Stokes (NS) equations were solved in this approach, and the flux at the interface was evaluated with a compressible lattice Boltzmann model. This method combined lattice Boltzmann method with finite volume method to solve Euler/NS equations. The proposed approach was validated by some simulations of one-dimensional and multi-dimensional problems.

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7

Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Development of Discrete Adjoint Approach Based on the Lattice Boltzmann Method." Advances in Mechanical Engineering 6 (January 1, 2014): 230854. http://dx.doi.org/10.1155/2014/230854.

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The purpose of this research is to present a general procedure with low implementation cost to develop the discrete adjoint approach for solving optimization problems based on the LB method. Initially, the macroscopic and microscopic discrete adjoint equations and the cost function gradient vector are derived mathematically, in detail, using the discrete LB equation. Meanwhile, for an elementary case, the analytical evaluation of the macroscopic and microscopic adjoint variables and the cost function gradients are presented. The investigation of the derivation procedure shows that the simplicity of the Boltzmann equation, as an alternative for the Navier-Stokes (NS) equations, can facilitate the process of extracting the discrete adjoint equation. Therefore, the implementation of the discrete adjoint equation based on the LB method needs fewer attempts than that of the NS equations. Finally, this approach is validated for the sample test case, and the results gained from the macroscopic and microscopic discrete adjoint equations are compared in an inverse optimization problem. The results show that the convergence rate of the optimization algorithm using both equations is identical and the evaluated gradients have a very good agreement with each other.

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8

Bernhoff, Niclas. "Boundary Layers and Shock Profiles for the Broadwell Model." International Journal of Differential Equations 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5801728.

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We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

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9

Banoo, K., F. Assad, and M. S. Lundstrom. "Formulation of the Boltzmann Equation as a Multi-Mode Drift-Diffusion Equation." VLSI Design 8, no. 1-4 (1998): 539–44. http://dx.doi.org/10.1155/1998/59373.

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We present a multi-mode drift-diffusion equation as reformulation of the Boltzmann equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

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10

BELLOUQUID, A. "A DIFFUSIVE LIMIT FOR NONLINEAR DISCRETE VELOCITY MODELS." Mathematical Models and Methods in Applied Sciences 13, no. 01 (2003): 35–58. http://dx.doi.org/10.1142/s0218202503002374.

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This paper is devoted to the analysis of the diffusive limit for a general discrete velocity model of the Boltzmann equation in any dimension of the physical space, for densities which remain in a suitable small neighbourhood of global Maxwellians. The scaled solutions of discrete Boltzmann equation are shown to have fluctuations that converge locally in time weakly to a limit governed by a solution of incompressible Stokes equations provided that the initial fluctuations are smooth. The weak limit becomes strong when the initial fluctuations converge to appropriate initial data. As applications the Carleman model and the one-dimensional Broadwell model are analyzed in detail.

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11

MARTYS, NICOS S. "ENERGY CONSERVING DISCRETE BOLTZMANN EQUATION FOR NONIDEAL SYSTEMS." International Journal of Modern Physics C 10, no. 07 (1999): 1367–82. http://dx.doi.org/10.1142/s0129183199001121.

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The BBGKY formalism is utilized to obtain a set of moment equations to be satisfied by the collision operator in an energy conserving discrete Boltzmann equation for the case of a nonlocal interaction potential. A modified BGK form of the collision operator consistent with these moment equations is described. In the regime of isothermal flows, a previous proposed nonideal gas model is recovered. Other approaches to constructing the collision operator are discussed. Numerical implementation of the modified BGK form, using a thermal lattice Boltzmann model, is illustrated as an example. The time dependence of the density autocorrelation function was studied for this model and found, at early times, to be strongly affected by the constraint of total energy conservation. The long time behavior of the density autocorrelation function was consistent with the theory of hydrodynamic fluctuations.

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12

NANBU, K. "MODEL KINETIC EQUATION FOR THE DISTRIBUTION OF DISCRETIZED INTERNAL ENERGY." Mathematical Models and Methods in Applied Sciences 04, no. 05 (1994): 669–75. http://dx.doi.org/10.1142/s0218202594000376.

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Kinetic equation for discretized internal energy is obtained by using the idea underlying the discrete-velocity kinetic theory. The equation satisfies the Boltzmann H-theorem. The solution of this equation in equilibrium is the Boltzmann distribution. The second moment of distribution shows an exponential relaxation.

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13

Fu, S. C., R. M. C. So, and W. W. F. Leung. "A Discrete Flux Scheme for Aerodynamic and Hydrodynamic Flows." Communications in Computational Physics 9, no. 5 (2011): 1257–83. http://dx.doi.org/10.4208/cicp.311009.241110s.

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AbstractThe objective of this paper is to seek an alternative to the numerical simulation of the Navier-Stokes equations by a method similar to solving the BGK-type modeled lattice Boltzmann equation. The proposed method is valid for both gas and liquid flows. A discrete flux scheme (DFS) is used to derive the governing equations for two distribution functions; one for mass and another for thermal energy. These equations are derived by considering an infinitesimally small control volume with a velocity lattice representation for the distribution functions. The zero-order moment equation of the mass distribution function is used to recover the continuity equation, while the first-order moment equation recovers the linear momentum equation. The recovered equations are correct to the first order of the Knudsen number(Kn);thus, satisfying the continuum assumption. Similarly, the zero-order moment equation of the thermal energy distribution function is used to recover the thermal energy equation. For aerodynamic flows, it is shown that the finite difference solution of the DFS is equivalent to solving the lattice Boltzmann equation (LBE) with a BGK-type model and a specified equation of state. Thus formulated, the DFS can be used to simulate a variety of aerodynamic and hydrodynamic flows. Examples of classical aeroacoustics, compressible flow with shocks, incompressible isothermal and non-isothermal Couette flows, stratified flow in a cavity, and double diffusive flow inside a rectangle are used to demonstrate the validity and extent of the DFS. Very good to excellent agreement with known analytical and/or numerical solutions is obtained; thus lending evidence to the DFS approach as an alternative to solving the Navier-Stokes equations for fluid flow simulations.

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14

He, Xiaoyi, Xiaowen Shan, and Gary D. Doolen. "Discrete Boltzmann equation model for nonideal gases." Physical Review E 57, no. 1 (1998): R13—R16. http://dx.doi.org/10.1103/physreve.57.r13.

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15

ANDALLAH, LAEK S., and HANS BABOVSKY. "A DISCRETE BOLTZMANN EQUATION BASED ON HEXAGONS." Mathematical Models and Methods in Applied Sciences 13, no. 11 (2003): 1537–63. http://dx.doi.org/10.1142/s0218202503003021.

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We develop the theory of a Boltzmann equation which is based on a hexagonal discretization of the velocity space. We prove that such a model contains all the basic features of classical kinetic theory, like collision invariants, H-theorem, equilibrium solutions, features of the linearized problem etc. This theory includes the infinite as well as finite hexagonal grids which may be used for numerical purposes.

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16

Makai, Mihály. "Discrete Symmetries of the Linear Boltzmann equation." Transport Theory and Statistical Physics 15, no. 3 (1986): 249–73. http://dx.doi.org/10.1080/00411458608210452.

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17

Cabannes, H. "The discrete model of the Boltzmann equation." Transport Theory and Statistical Physics 16, no. 4-6 (1987): 809–36. http://dx.doi.org/10.1080/00411458708204316.

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18

Junk, Michael, and Wen‐An Yong. "Rigorous Navier–Stokes limit of the lattice Boltzmann equation." Asymptotic Analysis 35, no. 2 (2003): 165–85. https://doi.org/10.3233/asy-2003-563.

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In this article, we rigorously investigate the diffusive limit of a velocity‐discrete Boltzmann equation which is used in the lattice Boltzmann method (LBM) to construct approximate solutions of the incompressible Navier–Stokes equation. Our results apply to LBM collision operators with multiple collision frequencies (generalized lattice Boltzmann) which include the widely used BGK operators.

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19

WANG, Y., Y. L. HE, Q. LI, G. H. TANG, and W. Q. TAO. "LATTICE BOLTZMANN MODEL FOR SIMULATING VISCOUS COMPRESSIBLE FLOWS." International Journal of Modern Physics C 21, no. 03 (2010): 383–407. http://dx.doi.org/10.1142/s0129183110015178.

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A lattice Boltzmann model is developed for viscous compressible flows with flexible specific-heat ratio and Prandtl number. Unlike the Maxwellian distribution function or circle function used in the existing lattice Boltzmann models, a polynomial kernel function in the phase space is introduced to recover the Navier–Stokes–Fourier equations. A discrete equilibrium density distribution function and a discrete equilibrium total energy distribution function are obtained from the discretization of the polynomial kernel function with Lagrangian interpolation. The equilibrium distribution functions are then coupled via the equation of state. In this framework, a model for viscous compressible flows is proposed. Several numerical tests from subsonic to supersonic flows, including the Sod shock tube, the double Mach reflection and the thermal Couette flow, are simulated to validate the present model. In particular, the discrete Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite-difference method. Numerical results agree well with the exact or analytic solutions. The present model has potential application in the study of complex fluid systems such as thermal compressible flows.

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20

Vedenyapin, Victor V., Sergey Z. Adzhiev, and Vladlena V. Kazantseva. "Entropy in the Sense of Boltzmann and Poincare, Boltzmann Extremals, and the Hamilton-Jacobi Method in Non-Hamiltonian Context." Contemporary Mathematics. Fundamental Directions 64, no. 1 (2018): 37–59. http://dx.doi.org/10.22363/2413-3639-2018-64-1-37-59.

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In this paper, we prove the H-theorem for generalized chemical kinetics equations. We consider important physical examples of such a generalization: discrete models of quantum kinetic equations (Uehling-Uhlenbeck equations) and a quantum Markov process (quantum random walk). We prove that time averages coincide with Boltzmann extremals for all such equations and for the Liouville equation as well. This gives us an approach for choosing the action-angle variables in the Hamilton-Jacobi method in a non-Hamiltonian context. We propose a simple derivation of the Hamilton-Jacobi equation from the Liouville equations in the ﬁnite-dimensional case.

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21

Marcos, Aboubacar, and Ambroise Soglo. "Solutions of a Class of Degenerate Kinetic Equations Using Steepest Descent in Wasserstein Space." Journal of Mathematics 2020 (June 9, 2020): 1–30. http://dx.doi.org/10.1155/2020/7489532.

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We use the steepest descent method in an Orlicz–Wasserstein space to study the existence of solutions for a very broad class of kinetic equations, which include the Boltzmann equation, the Vlasov–Poisson equation, the porous medium equation, and the parabolic p-Laplacian equation, among others. We combine a splitting technique along with an iterative variational scheme to build a discrete solution which converges to a weak solution of our problem.

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22

Vedenyapin, V. V., and S. A. Amosov. "Discrete models of the boltzmann equation for mixtures." Differential Equations 36, no. 7 (2000): 1027–32. http://dx.doi.org/10.1007/bf02754504.

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23

Koller, W., and F. Schürrer. "PNAPPROXIMATION OF THE NONLINEAR SEMI-DISCRETE BOLTZMANN EQUATION." Transport Theory and Statistical Physics 30, no. 4-6 (2001): 471–89. http://dx.doi.org/10.1081/tt-100105933.

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24

Vedenyapin, V. V., I. V. Mingalev, and O. V. Mingalev. "ON DISCRETE MODELS OF THE QUANTUM BOLTZMANN EQUATION." Russian Academy of Sciences. Sbornik Mathematics 80, no. 2 (1995): 271–85. http://dx.doi.org/10.1070/sm1995v080n02abeh003525.

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25

Preziosi, L. "Thermal creep problems by the discrete Boltzmann equation." Transport Theory and Statistical Physics 21, no. 3 (1992): 183–209. http://dx.doi.org/10.1080/00411459208203920.

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26

Shizuta, Yasushi, and Shuichi Kawashima. "The regular discrete models of the Boltzmann equation." Journal of Mathematics of Kyoto University 27, no. 1 (1987): 131–40. http://dx.doi.org/10.1215/kjm/1250520768.

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27

Lee, Taehun, and Ching-Long Lin. "A Characteristic Galerkin Method for Discrete Boltzmann Equation." Journal of Computational Physics 171, no. 1 (2001): 336–56. http://dx.doi.org/10.1006/jcph.2001.6791.

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28

Chen, Yu, Shulong Teng, Takauki Shukuwa, and Hirotada Ohashi. "Lattice-Boltzmann Simulation of Two-Phase Fluid Flows." International Journal of Modern Physics C 09, no. 08 (1998): 1383–91. http://dx.doi.org/10.1142/s0129183198001254.

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A model with a volumetric stress tensor added to the Navier–Stokes Equation is used to study two-phase fluid flows. The implementation of such an interface model into the lattice-Boltzmann equation is derived from the continuous Boltzmann BGK equation with an external force term, by using the discrete coordinate method. Numerical simulations are carried out for phase separation and "dam breaking" phenomena.

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29

Gu, Zhongyang, Xin Hu, Pritpal Matharu, Bartosz Protas, Makiko Sasada, and Tsuyoshi Yoneda. "The incompressible Navier–Stokes limit from the discrete-velocity BGK Boltzmann equation." Nonlinearity 38, no. 5 (2025): 055014. https://doi.org/10.1088/1361-6544/adca81.

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Abstract In this paper, we extend the Bardos–Golse–Levermore program (Bardos et al 1993 Commun. Pure Appl. Math. 46 667–753) to prove that a local weak solution to the d-dimensional incompressible Navier–Stokes equations ( d ⩾ 2 ) can be constructed by taking the hydrodynamic limit of a discrete-velocity Boltzmann equation with a simplified Bhatnagar–Gross–Krook collision operator. Moreover, in the case when the dimension is d = 2 , 3 , we characterise the combinations of finitely many particle velocities and probabilities that lead to the incompressible Navier–Stokes equations in the hydrodynamic limit. Numerical computations conducted in two-dimensional indicate that in the case of the simplest velocity lattice (D2Q9), the rate with which this hydrodynamic limit is achieved is of order O ( ε 2 ) , where ε → 0 is the Knudsen number. For the future investigations, it is worth considering if the hydrodynamic limit of the discrete-velocity Boltzmann equation can be also rigorously justified in the presence of non-trivial boundary conditions.

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30

Liu, Zhixiang, Chenkai Zhang, Wenhao Zhu, and Dongmei Huang. "A Physics-Informed Neural Network Based on the Boltzmann Equation with Multiple-Relaxation-Time Collision Operators." Axioms 13, no. 9 (2024): 588. http://dx.doi.org/10.3390/axioms13090588.

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The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at O10−3.

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31

Viggen, Erlend Magnus. "Sound Propagation Properties of the Discrete-Velocity Boltzmann Equation." Communications in Computational Physics 13, no. 3 (2013): 671–84. http://dx.doi.org/10.4208/cicp.271011.020212s.

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AbstractAs the numerical resolution is increased and the discretisation error decreases, the lattice Boltzmann method tends towards the discrete-velocity Boltzmann equation (DVBE). An expression for the propagation properties of plane sound waves is found for this equation. This expression is compared to similar ones from the Navier-Stokes and Burnett models, and is found to be closest to the latter. The anisotropy of sound propagation with the DVBE is examined using a two-dimensional velocity set. It is found that both the anisotropy and the deviation between the models is negligible if the Knudsen number is less than 1 by at least an order of magnitude.

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RINGHOFER, CHRISTIAN. "DISSIPATIVE DISCRETIZATION METHODS FOR APPROXIMATIONS TO THE BOLTZMANN EQUATION." Mathematical Models and Methods in Applied Sciences 11, no. 01 (2001): 133–48. http://dx.doi.org/10.1142/s0218202501000799.

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This paper deals with the spatial discretization of partial differential equations arising from Galerkin approximations to the Boltzmann equation, which preserves the entropy properties of the original collision operator. A general condition on finite difference methods is derived, which guarantees that the discrete system satisfies the appropriate equivalent of the entropy condition. As an application of this concept, entropy producing difference methods for the hydrodynamic model equations and for spherical harmonics expansions are presented.

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33

PREMNATH, KANNAN N., and JOHN ABRAHAM. "DISCRETE LATTICE BGK BOLTZMANN EQUATION COMPUTATIONS OF TRANSIENT INCOMPRESSIBLE TURBULENT JETS." International Journal of Modern Physics C 15, no. 05 (2004): 699–719. http://dx.doi.org/10.1142/s0129183104006157.

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In this paper, computations of transient, incompressible, turbulent, plane jets using the discrete lattice BGK Boltzmann equation are reported. Á priori derivation of the discrete lattice BGK Boltzmann equation with a spatially and temporally dependent relaxation time parameter, which is used to represent the averaged flow field, from its corresponding continuous form is given. The averaged behavior of the turbulence field is represented by the standard k–∊ turbulence model and computed using a finite-volume scheme on nonuniform grids. Computed results are compared with analytical solutions, experimental data and results of other computational methods. Satisfactory agreement is shown.

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SIEBERT, DIOGO NARDELLI, LUIZ ADOLFO HEGELE, RODRIGO SURMAS, LUÍS ORLANDO EMERICH DOS SANTOS, and PAULO CESAR PHILIPPI. "THERMAL LATTICE BOLTZMANN IN TWO DIMENSIONS." International Journal of Modern Physics C 18, no. 04 (2007): 546–55. http://dx.doi.org/10.1142/s0129183107010784.

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The velocity discretization is a critical step in deriving the lattice Boltzmann (LBE) from the Boltzmann equation. The velocity discretization problem was considered in a recent paper (Philippi et al., From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models, Physical Review E 73: 56702, 2006) following a new approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the LBE with respect to the Boltzmann equation. As a consequence, two-dimensional lattices and their respective equilibrium distributions were derived and discussed, considering the order of approximation that was required for the LBE. In the present work, a Chapman-Enskog (CE) analysis is performed for deriving the macroscopic transport equations for the mass, momentum and energy for these lattices. The problem of describing the transfer of energy in fluids is discussed in relation with the order of approximation of the LBE model. Simulation of temperature, pressure and velocity steps are also presented to validate the CE analysis.

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35

Zhai, Qinglan, Song Zheng, and Lin Zheng. "A kinetic theory based thermal lattice Boltzmann equation model." International Journal of Modern Physics C 28, no. 04 (2017): 1750047. http://dx.doi.org/10.1142/s0129183117500474.

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A thermal lattice Boltzmann equation (LBE) model within the framework of double distribution function (DDF) method is proposed from the continuous DDF Boltzmann equation, which has a clear physical significance. Since the discrete velocity set in present LBE model is not space filled, a Lax–Wendroff scheme is applied to solve the evolution equations by which the spatial interpolation of two distribution functions is overcome. To validate the model, some classical numerical tests include thermal Couette flow and natural convection flow are simulated, and the results agree well with the analytic solutions and other numerical results, which showed that the present model had the ability to describe the thermal fluid flow phenomena.

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36

Ilyin, Oleg. "Hybrid Lattice Boltzmann Model for Nonlinear Diffusion and Image Denoising." Mathematics 11, no. 22 (2023): 4601. http://dx.doi.org/10.3390/math11224601.

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In the present paper, a novel approach for image denoising based on the numerical solution to the nonlinear diffusion equation is proposed. The Perona–Malik-type equation is solved by employing a hybrid lattice Boltzmann model with five discrete velocities. In this method, the regions with large values of the diffusion coefficient are modeled with the lattice Boltzmann scheme for which hyper-viscous defects are reduced, while other regions are modeled with the conventional lattice Boltzmann model. The new method allows us to solve Perona–Malik-type equations with relatively large time steps and good accuracy. In numerical experiments, the removal of salt and pepper, speckle and Gaussian noise is considered. For salt and pepper noise, the novel scheme yields better peak signal-to-noise ratios in image denoising problems compared to the standard lattice Boltzmann approach. For certain non-small values of time steps, the novel model shows better results for speckle and Gaussian noise on average.

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37

Han, Linchang, Liming Yang, Zhihui Li, Jie Wu, Yinjie Du, and Xiang Shen. "Unlocking the Key to Accelerating Convergence in the Discrete Velocity Method for Flows in the Near Continuous/Continuous Flow Regimes." Entropy 25, no. 12 (2023): 1609. http://dx.doi.org/10.3390/e25121609.

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How to improve the computational efficiency of flow field simulations around irregular objects in near-continuum and continuum flow regimes has always been a challenge in the aerospace re-entry process. The discrete velocity method (DVM) is a commonly used algorithm for the discretized solutions of the Boltzmann-BGK model equation. However, the discretization of both physical and molecular velocity spaces in DVM can result in significant computational costs. This paper focuses on unlocking the key to accelerate the convergence in DVM calculations, thereby reducing the computational burden. Three versions of DVM are investigated: the semi-implicit DVM (DVM-I), fully implicit DVM (DVM-II), and fully implicit DVM with an inner iteration of the macroscopic governing equation (DVM-III). In order to achieve full implicit discretization of the collision term in the Boltzmann-BGK equation, it is necessary to solve the corresponding macroscopic governing equation in DVM-II and DVM-III. In DVM-III, an inner iterative process of the macroscopic governing equation is employed between two adjacent DVM steps, enabling a more accurate prediction of the equilibrium state for the full implicit discretization of the collision term. Fortunately, the computational cost of solving the macroscopic governing equation is significantly lower than that of the Boltzmann-BGK equation. This is primarily due to the smaller number of conservative variables in the macroscopic governing equation compared to the discrete velocity distribution functions in the Boltzmann-BGK equation. Our findings demonstrate that the fully implicit discretization of the collision term in the Boltzmann-BGK equation can accelerate DVM calculations by one order of magnitude in continuum and near-continuum flow regimes. Furthermore, the introduction of the inner iteration of the macroscopic governing equation provides an additional 1–2 orders of magnitude acceleration. Such advancements hold promise in providing a computational approach for simulating flows around irregular objects in near-space environments.

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Ren, Junjie, Ping Guo, and Zhaoli Guo. "Rectangular Lattice Boltzmann Equation for Gaseous Microscale Flow." Advances in Applied Mathematics and Mechanics 8, no. 2 (2014): 306–30. http://dx.doi.org/10.4208/aamm.2014.m672.

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AbstractThe lattice Boltzmann equation (LBE) is considered as a promising approach for simulating flows of liquid and gas. Most of LBE studies have been devoted to regular square LBE and few works have focused on the rectangular LBE in the simulation of gaseous microscale flows. In fact, the rectangular LBE, as an alternative and efficient method, has some advantages over the square LBE in simulating flows with certain computational domains of large aspect ratio (e.g., long micro channels). Therefore, in this paper we expand the application scopes of the rectangular LBE to gaseous microscale flow. The kinetic boundary conditions for the rectangular LBE with a multiple-relaxation-time (MRT) collision operator, i.e., the combined bounce-back/specular-reflection (CBBSR) boundary condition and the discrete Maxwell's diffuse-reflection (DMDR) boundary condition, are studied in detail. We observe some discrete effects in both the CBBSR and DMDR boundary conditions for the rectangular LBE and present a reasonable approach to overcome these discrete effects in the two boundary conditions. It is found that the DMDR boundary condition for the square MRT-LBE can not realize the real fully diffusive boundary condition, while the DMDR boundary condition for the rectangular MRT-LBE with the grid aspect ratio a≠1 can do it well. Some numerical tests are implemented to validate the presented theoretical analysis. In addition, the computational efficiency and relative difference between the rectangular LBE and the square LBE are analyzed in detail. The rectangular LBE is found to be an efficient method for simulating the gaseous microscale flows in domains with large aspect ratios.

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39

Shizuta, Yasushi, and Shuichi Kawashima. "The regularity of discrete models of the Boltzmann equation." Proceedings of the Japan Academy, Series A, Mathematical Sciences 61, no. 8 (1985): 252–54. http://dx.doi.org/10.3792/pjaa.61.252.

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40

Gross, M., M. E. Cates, F. Varnik, and R. Adhikari. "Langevin theory of fluctuations in the discrete Boltzmann equation." Journal of Statistical Mechanics: Theory and Experiment 2011, no. 03 (2011): P03030. http://dx.doi.org/10.1088/1742-5468/2011/03/p03030.

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41

Amossov, Stepan A. "Discrete kinetic models of relativistic Boltzmann equation for mixtures." Physica A: Statistical Mechanics and its Applications 301, no. 1-4 (2001): 330–40. http://dx.doi.org/10.1016/s0378-4371(01)00380-6.

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Płatkowski, T., and W. Waluś. "An efficient discrete-velocity method for the Boltzmann equation." Computer Physics Communications 121-122 (September 1999): 717. http://dx.doi.org/10.1016/s0010-4655(06)70120-5.

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43

La Rocca, Michele, Andrea Montessori, Pietro Prestininzi, and Lakshmanan Elango. "Discrete Boltzmann Equation model of polydisperse shallow granular flows." International Journal of Multiphase Flow 113 (April 2019): 107–16. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2019.01.008.

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44

Wagner, Wolfgang. "Approximation of the Boltzmann equation by discrete velocity models." Journal of Statistical Physics 78, no. 5-6 (1995): 1555–70. http://dx.doi.org/10.1007/bf02180142.

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45

Kawashima, Shuichi. "Asymptotic stability of Maxwellians of the discrete Boltzmann equation." Transport Theory and Statistical Physics 16, no. 4-6 (1987): 781–93. http://dx.doi.org/10.1080/00411458708204314.

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46

Mischler, Stéphane. "Convergence of Discrete-Velocity Schemes for the Boltzmann Equation." Archive for Rational Mechanics and Analysis 140, no. 1 (1997): 53–77. http://dx.doi.org/10.1007/s002050050060.

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47

Zhang, Zhenyu, Wei Zhao, Qingjun Zhao, Guojing Lu, and Jianzhong Xu. "Inlet and outlet boundary conditions for the discrete velocity direction model." Modern Physics Letters B 32, no. 04 (2018): 1850048. http://dx.doi.org/10.1142/s0217984918500483.

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The discrete velocity direction model is an approximate method to the Boltzmann equation, which is an optional kinetic method to microgas flow and heat transfer. In this paper, the treatment of the inlet and outlet boundary conditions for the model is proposed. In the computation strategy, the microscopic molecular speed distribution functions at inlet and outlet are indirectly determined by the macroscopic gas pressure, mass flux and temperature, which are all measurable parameters in microgas flow and heat transfer. The discrete velocity direction model with the pressure correction boundary conditions was applied into the plane Poiseuille flow in microscales and the calculations cover all flow regimes. The numerical results agree well with the data of the NS equation near the continuum regime and the date of linearized Boltzmann equation and the DSMC method in the transition regime and free molecular flow. The Knudsen paradox and the nonlinear pressure distributions have been accurately captured by the discrete velocity direction model with the present boundary conditions.

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48

Boghosian, Bruce M., Luis M. Fazendeiro, Jonas Lätt, Hui Tang, and Peter V. Coveney. "New variational principles for locating periodic orbits of differential equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1944 (2011): 2211–18. http://dx.doi.org/10.1098/rsta.2011.0066.

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We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier–Stokes equations, simulated using the lattice Boltzmann equation.

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Premnath, Kannan N., and Sanjoy Banerjee. "Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation: Implications for Filtered Turbulence Simulation." Communications in Computational Physics 12, no. 3 (2012): 732–66. http://dx.doi.org/10.4208/cicp.181210.090911a.

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AbstractWe present a systematic derivation of a model based on the central moment lattice Boltzmann equation that rigorously maintains Galilean invariance of forces to simulate inertial frame independent flow fields. In this regard, the central moments, i.e. moments shifted by the local fluid velocity, of the discrete source terms of the lattice Boltzmann equation are obtained by matching those of the continuous full Boltzmann equation of various orders. This results in an exact hierarchical identity between the central moments of the source terms of a given order and the components of the central moments of the distribution functions and sources of lower orders. The corresponding source terms in velocity space are then obtained from an exact inverse transformation due to a suitable choice of orthogonal basis for moments. Furthermore, such a central moment based kinetic model is further extended by incorporating reduced compressibility effects to represent incompressible flow. Moreover, the description and simulation of fluid turbulence for full or any subset of scales or their averaged behavior should remain independent of any inertial frame of reference. Thus, based on the above formulation, a new approach in lattice Boltzmann framework to incorporate turbulence models for simulation of Galilean invariant statistical averaged or filtered turbulent fluid motion is discussed.

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Buet, C. "Conservative and Entropy Schemes for Boltzmann Collision Operator of Polyatomic Gases." Mathematical Models and Methods in Applied Sciences 07, no. 02 (1997): 165–92. http://dx.doi.org/10.1142/s0218202597000116.

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We propose two discrete velocity models derived from the Boltzmann equation of Larsen–Borgnakke type for polyatomic gases. These two models are natural extensions of previously discussed discrete velocity models used for monoatomic gases. These two models have the same properties as the continuous one, which are conservation of mass, momentum and energy, discrete Maxwellians as equilibrium states and H-theorems.

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