Artykuły w czasopismach: „Discrete Velocity Boltzmann Schemes”

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Hsu, C. T., S. W. Chiang, and K. F. Sin. "A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods." Communications in Computational Physics 11, no. 4 (2012): 1397–414. http://dx.doi.org/10.4208/cicp.150510.150511s.

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AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

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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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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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Diaz, Manuel A., Min-Hung Chen, and Jaw-Yen Yang. "High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation." Communications in Computational Physics 18, no. 4 (2015): 1012–49. http://dx.doi.org/10.4208/cicp.171214.210715s.

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AbstractHigh-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

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MATTILA, KEIJO K., DIOGO N. SIEBERT, LUIZ A. HEGELE, and PAULO C. PHILIPPI. "HIGH-ORDER LATTICE-BOLTZMANN EQUATIONS AND STENCILS FOR MULTIPHASE MODELS." International Journal of Modern Physics C 24, no. 12 (2013): 1340006. http://dx.doi.org/10.1142/s0129183113400068.

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The lattice Boltzmann (LB) method, based on mesoscopic modeling of transport phenomena, appears to be an attractive alternative for the simulation of complex fluid flows. Examples of such complex dynamics are multiphase and multicomponent flows for which several LB models have already been proposed. However, due to theoretical or numerical reasons, some of these models may require application of high-order lattice-Boltzmann equations (LBEs) and stencils. Here, we will present a derivation of LBEs from the discrete-velocity Boltzmann equation (DVBE). By using the method of characteristics, high-order accurate equations are conveniently formulated with standard numerical methods for ordinary differential equations (ODEs). In particular, we will derive implicit LB schemes due to their stability properties. A simple algorithm is presented which enables implementation of the implicit schemes without resorting to, e.g. change of variables. Finally, some numerical experiments with high-order equations and stencils together with two specific multiphase models are presented.

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Wang, Liang, Xuhui Meng, Hao-Chi Wu, Tian-Hu Wang, and Gui Lu. "Discrete effect on single-node boundary schemes of lattice Bhatnagar–Gross–Krook model for convection-diffusion equations." International Journal of Modern Physics C 31, no. 01 (2019): 2050017. http://dx.doi.org/10.1142/s0129183120500175.

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The discrete effect on the boundary condition has been a fundamental topic for the lattice Boltzmann method (LBM) in simulating heat and mass transfer problems. In previous works based on the anti-bounce-back (ABB) boundary condition for convection-diffusion equations (CDEs), it is indicated that the discrete effect cannot be commonly removed in the Bhatnagar–Gross–Krook (BGK) model except for a special value of relaxation time. Targeting this point in this paper, we still proceed within the framework of BGK model for two-dimensional CDEs, and analyze the discrete effect on a non-halfway single-node boundary condition which incorporates the effect of the distance ratio. By analyzing an unidirectional diffusion problem with a parabolic distribution, the theoretical derivations with three different discrete velocity models show that the numerical slip is a combined function of the relaxation time and the distance ratio. Different from previous works, we definitely find that the relaxation time can be freely adjusted by the distance ratio in a proper range to eliminate the numerical slip. Some numerical simulations are carried out to validate the theoretical derivations, and the numerical results for the cases of straight and curved boundaries confirm our theoretical analysis. Finally, it should be noted that the present analysis can be extended from the BGK model to other lattice Boltzmann (LB) collision models for CDEs, which can broaden the parameter range of the relaxation time to approach 0.5.

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Mieussens, Luc. "Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries." Journal of Computational Physics 162, no. 2 (2000): 429–66. http://dx.doi.org/10.1006/jcph.2000.6548.

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Aristov, V. V., O. V. Ilyin, and O. A. Rogozin. "Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods." Journal of Computational Science 40 (February 2020): 101064. http://dx.doi.org/10.1016/j.jocs.2019.101064.

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Buet, C. "A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics." Transport Theory and Statistical Physics 25, no. 1 (1996): 33–60. http://dx.doi.org/10.1080/00411459608204829.

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Wu, Junlin, Zhihui Li, Aoping Peng, and Xinyu Jiang. "Numerical Simulations of Unsteady Flows From Rarefied Transition to Continuum Using Gas-Kinetic Unified Algorithm." Advances in Applied Mathematics and Mechanics 7, no. 5 (2015): 569–96. http://dx.doi.org/10.4208/aamm.2014.m523.

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AbstractNumerical simulations of unsteady gas flows are studied on the basis of Gas-Kinetic Unified Algorithm (GKUA) from rarefied transition to continuum flow regimes. Several typical examples are adopted. An unsteady flow solver is developed by solving the Boltzmann model equations, including the Shakhov model and the Rykov model etc. The Rykov kinetic equation involving the effect of rotational energy can be transformed into two kinetic governing equations with inelastic and elastic collisions by integrating the molecular velocity distribution function with the weight factor on the energy of rotational motion. Then, the reduced velocity distribution functions are devised to further simplify the governing equation for one- and two-dimensional flows. The simultaneous equations are numerically solved by the discrete velocity ordinate (DVO) method in velocity space and the finite-difference schemes in physical space. The time-explicit operator-splitting scheme is constructed, and numerical stability conditions to ascertain the time step are discussed. As the application of the newly developed GKUA, several unsteady varying processes of one- and two-dimensional flows with different Knudsen number are simulated, and the unsteady transport phenomena and rarefied effects are revealed and analyzed. It is validated that the GKUA solver is competent for simulations of unsteady gas dynamics covering various flow regimes.

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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 (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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Yang, Jaw-Yen, Bagus Putra Muljadi, Zhi-Hui Li, and Han-Xin Zhang. "A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics." Communications in Computational Physics 14, no. 1 (2013): 242–64. http://dx.doi.org/10.4208/cicp.290112.030812a.

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AbstractAn accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

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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 (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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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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Sun, Yifang, Sen Zou, Guang Zhao, and Bei Yang. "THE IMPROVEMENT AND REALIZATION OF FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD." Aerospace technic and technology, no. 1 (February 26, 2021): 4–13. http://dx.doi.org/10.32620/aktt.2021.1.01.

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The Lattice Boltzmann Method (LBM) is a numerical method developed in recent decades. It has the characteristics of high parallel efficiency and simple boundary processing. The basic idea is to construct a simplified dynamic model so that the macroscopic behavior of the model is the same as the macroscopic equation. From the perspective of micro-dynamics, LBM treats macro-physical quantities as micro-quantities to obtain results by statistical averaging. The Finite-difference LBM (FDLBM) is a new numerical method developed based on LBM. The first finite-difference LBE (FDLBE) was perhaps due to Tamura and Akinori and was examined by Cao et al. in more detail. Finite-difference LBM was further extended to curvilinear coordinates with nonuniform grids by Mei and Shyy. By improving the FDLBE proposed by Mei and Shyy, a new finite difference LBM is obtained in the paper. In the model, the collision term is treated implicitly, just as done in the Mei-Shyy model. However, by introducing another distribution function based on the earlier distribution function， the implicitness of the discrete scheme is eliminated, and a simple explicit scheme is finally obtained, such as the standard LBE. Furthermore, this trick for the FDLBE can also be easily used to develop more efficient FVLBE and FELBE schemes. To verify the correctness and feasibility of this improved FDLBM model, which is used to calculate the square cavity model, and the calculated results are compared with the data of the classic square cavity model. The comparison result includes two items: the velocity on the centerline of the square cavity and the position of the vortex center in the square cavity. The simulation results of FDLBM are very consistent with the data in the literature. When Re=400, the velocity profiles of u and v on the centerline of the square cavity are consistent with the data results in Ghia's paper, and the vortex center position in the square cavity is also almost the same as the data results in Ghia's paper. Therefore, the verification of FDLBM is successful and FDLBM is feasible. This improved method can also serve as a reference for subsequent research.

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Gan, Yanbiao, Aiguo Xu, Guangcai Zhang, Junqi Wang, Xijun Yu, and Yang Yang. "Lattice Boltzmann kinetic modeling and simulation of thermal liquid–vapor system." International Journal of Modern Physics C 25, no. 12 (2014): 1441002. http://dx.doi.org/10.1142/s0129183114410022.

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We present a highly efficient lattice Boltzmann (LB) kinetic model for thermal liquid–vapor system. Three key components are as below: (i) a discrete velocity model (DVM) by Kataoka et al. [Phys. Rev. E69, 035701(R) (2004)]; (ii) a forcing term Ii aiming to describe the interfacial stress and recover the van der Waals (VDW) equation of state (EOS) by Gonnella et al. [Phys. Rev. E76, 036703 (2007)] and (iii) a Windowed Fast Fourier Transform (WFFT) scheme and its inverse by our group [Phys. Rev. E84, 046715 (2011)] for solving the spatial derivatives, together with a second-order Runge–Kutta (RK) finite difference scheme for solving the temporal derivative in the LB equation. The model is verified and validated by well-known benchmark tests. The results recovered from the present model are well consistent with previous ones [Phys. Rev. E84, 046715 (2011)] or theoretical analysis. The usage of less discrete velocities, high-order RK algorithm and WFFT scheme with 16th-order in precision makes the model more efficient by about 10 times and more accurate than the original one.

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Klar, Axel. "Relaxation Scheme for a Lattice–Boltzmann-type Discrete Velocity Model and Numerical Navier–Stokes Limit." Journal of Computational Physics 148, no. 2 (1999): 416–32. http://dx.doi.org/10.1006/jcph.1998.6123.

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Xu, Lei, Wu Zhang, Zhengzheng Yan, Zheng Du, and Rongliang Chen. "A novel median dual finite volume lattice Boltzmann method for incompressible flows on unstructured grids." International Journal of Modern Physics C 31, no. 12 (2020): 2050173. http://dx.doi.org/10.1142/s0129183120501739.

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A novel median dual finite volume lattice Boltzmann method (FV-LBM) for the accurate simulation of incompressible flows on unstructured grids is presented in this paper. The finite volume method is adopted to discretize the discrete velocity Boltzmann equation (DVBE) on median dual control volumes (CVs). In the previous studies on median dual FV-LBMs, the fluxes for each partial face have to be computed separately. In the present second-order scheme, we assume the particle distribution functions (PDFs) to be constant for all faces grouped around a particular edge. The fluxes are then evaluated using the low-diffusion Roe scheme at the midpoint of the edge, and the PDFs at the faces of the CV are obtained through piecewise linear reconstruction of the left and right states. The gradients of the PDFs are computed with the Green–Gauss approach. The presented scheme is validated on four benchmark flows: (a) pressure driven Poiseuille flow; (b) the backward-facing step flow with [Formula: see text], 100, 200 and 300; (c) the lid-driven flow with [Formula: see text] and 1000; and (d) the steady viscous flow past a circular cylinder with [Formula: see text], 20 and 40.

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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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Watanabe, Seiya, Changhong Hu, and Takayuki Aoki. "Coupled Lattice Boltzmann and Discrete Element Simulations of Ship-Ice Interactions." IOP Conference Series: Materials Science and Engineering 1288, no. 1 (2023): 012015. http://dx.doi.org/10.1088/1757-899x/1288/1/012015.

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Abstract Evaluating ice loads acting on ships is essential for the safety of ships navigating in ice-covered seas. In this study, we develop a CFD method to handle ship, ice, and fluid interaction. The lattice Boltzmann method, capable of large-scale calculations, is applied to the simulation of free-surface fluids. The ice motion is computed by solving the equations of motion of a rigid body, and the discrete element method models the ice-ice and ice-ship contact interactions. A momentum exchange scheme couples the lattice Boltzmann method and particle-based rigid body simulation. We introduce tree-based adaptive mesh refinement and multiple GPU computing to improve grid resolution and computational time. The proposed method is applied to model scale simulations of ship navigation in a brash ice channel. Simulations were performed for various conditions with different ice concentrations and ship velocities, and we observed that ice resistance increased with the ice concentration and the ship velocity increased. The ice motions and resistances obtained from our simulations are reasonable compared to model equations of Finnish-Swedish ice class rules (FSICR) and numerical analyses of a previous study.

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Krivovichev, Gerasim V., and Elena S. Bezrukova. "Analysis of Discrete Velocity Models for Lattice Boltzmann Simulations of Compressible Flows at Arbitrary Specific Heat Ratio." Computation 11, no. 7 (2023): 138. http://dx.doi.org/10.3390/computation11070138.

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This paper is devoted to the comparison of discrete velocity models used for simulation of compressible flows with arbitrary specific heat ratios in the lattice Boltzmann method. The stability of the governing equations is analyzed for the steady flow regime. A technique for the construction of stability domains in parametric space based on the analysis of eigenvalues is proposed. A comparison of stability domains for different models is performed. It is demonstrated that the maximum value of macrovelocity, which defines instability initiation, is dependent on the values of relaxation time, and plots of this dependence are constructed. For double-distribution-function models, it is demonstrated that the value of the Prantdl number does not seriously affect stability. The off-lattice parametric finite-difference scheme is proposed for the practical realization of the considered kinetic models. The Riemann problems and the problem of Kelvin–Helmholtz instability simulation are numerically solved. It is demonstrated that different models lead to close numerical results. The proposed technique of stability investigation can be used as an effective tool for the theoretical comparison of different kinetic models used in applications of the lattice Boltzmann method.

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Liu, Bowen, and Weiping Shi. "A Non-Equilibrium Interpolation Scheme for IB-LBM Optimized by Approximate Force." Axioms 12, no. 3 (2023): 298. http://dx.doi.org/10.3390/axioms12030298.

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A non-equilibrium scheme and an optimized approximate force are proposed for the immersed boundary–lattice Boltzmann method (IB-LBM) to solve the fluid–structure interaction (FSI) equations. This new IB-LBM uses the discrete velocity distribution function and non-equilibrium distribution function to establish the interpolation operator and the spread operator at the mesoscopic scale. In the interpolation operator, we use the force model of LBM to derive a direct force with a simple form. In the spread operator, we give a theoretical proof with local second-order accuracy of the spread process using the non-equilibrium theory from the LBM. A non-iterative explicit force approximation scheme optimizes the direct force in that the streamlines have no penetration phenomenon, and the no-slip condition is strictly satisfied. Different from other schemes for the IB-LBM, we try to apply the non-equilibrium theory from the LBM to the IB-LBM and obtain good results. The explicit force obtained using the non-equilibrium scheme and then optimized via the non-iterative streamline correction equation simplifies the explicit direct force scheme and the original implicit scheme previously proposed but obtains a similar streamline correction result compared with the implicit method. Numerical tests prove the applicability and accuracy of this method in the simulation of complex conditions such as moving rigid bodies and deforming flexible bodies.

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Guo, Wenqiang, and Guoxiang Hou. "Three-Dimensional Simulations of Anisotropic Slip Microflows Using the Discrete Unified Gas Kinetic Scheme." Entropy 24, no. 7 (2022): 907. http://dx.doi.org/10.3390/e24070907.

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The specific objective of the present work study is to propose an anisotropic slip boundary condition for three-dimensional (3D) simulations with adjustable streamwise and spanwise slip length by the discrete unified gas kinetic scheme (DUGKS). The present boundary condition is proposed based on the assumption of nonlinear velocity profiles near the wall instead of linear velocity profiles in a unidirectional steady flow. Moreover, a 3D corner boundary condition is introduced to the DUGKS to reduce the singularities. Numerical tests validate the effectiveness of the present method, which is more accurate than the bounce-back and specular reflection slip boundary condition in the lattice Boltzmann method. It is of significance to study the lid-driven cavity flow due to its applications and its capability in exhibiting important phenomena. Then, the present work explores, for the first time, the effects of anisotropic slip on the two-sided orthogonal oscillating micro-lid-driven cavity flow by adopting the present method. This work will generate fresh insight into the effects of anisotropic slip on the 3D flow in a two-sided orthogonal oscillating micro-lid-driven cavity. Some findings are obtained: The oscillating velocity of the wall has a weaker influence on the normal velocity component than on the tangential velocity component. In most cases, large slip length has a more significant influence on velocity profiles than small slip length. Compared with pure slip in both top and bottom walls, anisotropic slip on the top wall has a greater influence on flow, increasing the 3D mixing of flow. In short, the influence of slip on the flow field depends not only on slip length but also on the relative direction of the wall motion and the slip velocity. The findings can help in better understanding the anisotropic slip effect on the unsteady microflow and the design of microdevices.

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Koellermeier, Julian, and Manuel Torrilhon. "Numerical Study of Partially Conservative Moment Equations in Kinetic Theory." Communications in Computational Physics 21, no. 4 (2017): 981–1011. http://dx.doi.org/10.4208/cicp.oa-2016-0053.

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AbstractMoment models are often used for the solution of kinetic equations such as the Boltzmann equation. Unfortunately, standard models like Grad's equations are not hyperbolic and can lead to nonphysical solutions. Newly derived moment models like the Hyperbolic Moment Equations and the Quadrature-Based Moment Equations yield globally hyperbolic equations but are given in partially conservative form that cannot be written as a conservative system.In this paper we investigate the applicability of different dedicated numerical schemes to solve the partially conservative model equations. Caused by the non-conservative type of equation we obtain differences in the numerical solutions, but due to the structure of the moment systems we show that these effects are very small for standard simulation cases. After successful identification of useful numerical settings we show a convergence study for a shock tube problem and compare the results to a discrete velocity solution. The results are in good agreement with the reference solution and we see convergence considering an increasing number of moments.

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Varmazyar, Mostafa, and Majid Bazargan. "Generalized Coordinate Transformation for Lattice Boltzmann Equation Using TTM Structured Grid Generation." Advanced Materials Research 433-440 (January 2012): 3371–77. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.3371.

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The standard lattice Boltzmann method has only been applied to the uniform structured grid so far. This limitation can be removed by discretization of the position as well as the velocity space separately. In the present study the generalized coordinate is introduced to transform the lattice Boltzmann Equation (LBE) from physical domain to computational domain. This method uses the finite difference method to discrete the local derivatives in computational space. The central scheme uses to estimate the convection term. The generalized coordinate transformation method introduced in this study has been validated against a sample case study. For this purpose, an unsteady Couette flow between two cylinders has been examined. Good agreement between most of the results of this study and the available data in the literature is reached. The source of some discrepancies between the current results and available data seems to be due to discretization method used in present work to calculate the Jacobian and metrics of transformation matrix.

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Goodarzi, M., M. R. Safaei, A. Karimipour, et al. "Comparison of the Finite Volume and Lattice Boltzmann Methods for Solving Natural Convection Heat Transfer Problems inside Cavities and Enclosures." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/762184.

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Different numerical methods have been implemented to simulate internal natural convection heat transfer and also to identify the most accurate and efficient one. A laterally heated square enclosure, filled with air, was studied. A FORTRAN code based on the lattice Boltzmann method (LBM) was developed for this purpose. The finite difference method was applied to discretize the LBM equations. Furthermore, for comparison purpose, the commercially available CFD package FLUENT, which uses finite volume Method (FVM), was also used to simulate the same problem. Different discretization schemes, being the first order upwind, second order upwind, power law, and QUICK, were used with the finite volume solver where the SIMPLE and SIMPLEC algorithms linked the velocity-pressure terms. The results were also compared with existing experimental and numerical data. It was observed that the finite volume method requires less CPU usage time and yields more accurate results compared to the LBM. It has been noted that the 1st order upwind/SIMPLEC combination converges comparatively quickly with a very high accuracy especially at the boundaries. Interestingly, all variants of FVM discretization/pressure-velocity linking methods lead to almost the same number of iterations to converge but higher-order schemes ask for longer iterations.

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Cheng, Yongguang, Luoding Zhu, and Chunze Zhang. "Numerical Study of Stability and Accuracy of the Immersed Boundary Method Coupled to the Lattice Boltzmann BGK Model." Communications in Computational Physics 16, no. 1 (2014): 136–68. http://dx.doi.org/10.4208/cicp.260313.291113a.

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AbstractThis paper aims to study the numerical features of a coupling scheme between the immersed boundary (IB) method and the lattice Boltzmann BGK (LBGK) model by four typical test problems: the relaxation of a circular membrane, the shearing flow induced by a moving fiber in the middle of a channel, the shearing flow near a non-slip rigid wall, and the circular Couette flow between two inversely rotating cylinders. The accuracy and robustness of the IB-LBGK coupling scheme, the performances of different discrete Dirac delta functions, the effect of iteration on the coupling scheme, the importance of the external forcing term treatment, the sensitivity of the coupling scheme to flow and boundary parameters, the velocity slip near non-slip rigid wall, and the origination of numerical instabilities are investigated in detail via the four test cases. It is found that the iteration in the coupling cycle can effectively improve stability, the introduction of a second-order forcing term in LBGK model is crucial, the discrete fiber segment length and the orientation of the fiber boundary obviously affect accuracy and stability, and the emergence of both temporal and spatial fluctuations of boundary parameters seems to be the indication of numerical instability. These elaborate results shed light on the nature of the coupling scheme and may benefit those who wish to use or improve the method.

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Moufekkir, F., M. A. Moussaoui, A. Mezrhab, and H. Naji. "Computation of coupled double-diffusive convection–radiation including lattice Boltzmann simulation of fluid flow." Journal of Fluid Mechanics 728 (July 3, 2013): 146–62. http://dx.doi.org/10.1017/jfm.2013.282.

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AbstractThis paper reports a numerical study of coupled double diffusive convection and radiation in a differentially heated square enclosure filled with non-grey air–CO2 (or air–H2O) mixtures. The numerical procedure is based on a hybrid scheme with the multiple relaxation time lattice Boltzmann method and the finite difference method. The fluid velocity is determined by the D2Q9 multiple relaxation time model, and the energy equation is discretized by the finite difference method to compute the temperature field, while the radiative part of the energy equation is calculated by the discrete ordinates method combined with the spectral line-based weighted sum of grey gases model. Depending on the boundary conditions, aiding and opposing flows occur as the result of temperature and concentration gradients. The effects of various parameters, such as the molar fraction on the flow structure, thermal and concentration fields, are investigated for aiding and opposing cases. The numerical results show that, in the presence of non-grey radiation, the heat transfer is decreased and the mass transfer is slightly modified. The gas radiation modifies the structure of the velocity and thermal fields by generating inclined stratifications and promoting instabilities in opposing flows.

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Wang, Peng, Lianhua Zhu, Zhaoli Guo, and Kun Xu. "A Comparative Study of LBE and DUGKS Methods for Nearly Incompressible Flows." Communications in Computational Physics 17, no. 3 (2015): 657–81. http://dx.doi.org/10.4208/cicp.240614.171014a.

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AbstractThe lattice Boltzmann equation (LBE) methods (both LBGK and MRT) and the discrete unified gas-kinetic scheme (DUGKS) are both derived from the Boltzmann equation, but with different consideration in their algorithm construction. With the same numerical discretization in the particle velocity space, the distinctive modeling of these methods in the update of gas distribution function may introduce differences in the computational results. In order to quantitatively evaluate the performance of these methods in terms of accuracy, stability, and efficiency, in this paper we test LBGK, MRT, and DUGKS in two-dimensional cavity flow and the flow over a square cylinder, respectively. The results for both cases are validated against benchmark solutions. The numerical comparison shows that, with sufficient mesh resolution, the LBE and DUGKS methods yield qualitatively similar results in both test cases. With identical mesh resolutions in both physical and particle velocity space, the LBE methods are more efficient than the DUGKS due to the additional particle collision modeling in DUGKS. But, the DUGKS is more robust and accurate than the LBE methods in most test conditions. Particularly, for the unsteady flow over a square cylinder at Reynolds number 300, with the same mesh resolution it is surprisingly observed that the DUGKS can capture the physical multi-frequency vortex shedding phenomena while the LBGK and MRT fail to get that. Furthermore, the DUGKS is a finite volume method and its computational efficiency can be much improved when a non-uniform mesh in the physical space is adopted. The comparison in this paper clearly demonstrates the progressive improvement of the lattice Boltzmann methods from LBGK, to MRT, up to the current DUGKS, along with the inclusion of more reliable physical process in their algorithm development. Besides presenting the Navier-Stokes solution, the DUGKS can capture the rarefied flow phenomena as well with the increasing of Knudsen number.

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Wu, Jun-Lin, Zhi-Hui Li, Ao-Ping Peng, Xing-Cai Pi, and Xin-Yu Jiang. "Utility computable modeling of a Boltzmann model equation for bimolecular chemical reactions and numerical application." Physics of Fluids 34, no. 4 (2022): 046111. http://dx.doi.org/10.1063/5.0088440.

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A Boltzmann model equation (kinetic model) involving the chemical reaction of a multicomponent gaseous mixture is derived based on Groppi's work [“A Bhatnagar–Gross–Krook-type approach for chemically reacting gas mixtures,” Phys. Fluids 16, 4273 (2004)], in which the relaxation parameters of elastic collision frequency for rigid elastic spheres are obtained based on the collision term, and the pivotal collision frequency of the chemical reaction is deduced from the chemical reaction rate that is determined by the direct simulation Monte Carlo (DSMC) method. This kinetic model is shown to be conservative, and the H theorem for an endothermic reaction is proven. In the framework of the gas-kinetic unified algorithm, the discrete velocity method, finite volume method, and implicit scheme are applied to solve the proposed kinetic model by introducing a suitable boundary condition at the wall surface. For hypersonic flows around a cylinder, the proposed kinetic model and the corresponding numerical methods are verified for both endothermic and exothermic reactions by comparison of the model's results with results from the DSMC method. The different influences of endothermic and exothermic reactions are also given. Finally, the proposed kinetic model is also used to simulate an exothermic reaction-driven flow in a square cavity.

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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 (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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Su, Yan, Tiniao Ng, Yinping Zhang, and Jane H. Davidson. "Three dimensional thermal diffusion in anisotropic heterogeneous structures simulated by a non-dimensional lattice Boltzmann method with a controllable structure generation scheme based on discrete Gaussian quadrature space and velocity." International Journal of Heat and Mass Transfer 108 (May 2017): 386–401. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.023.

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Зипунова, Елизавета Вячеславовна, Анастасия Юрьевна Перепёлкина, and Андрей Владимирович Закиров. "Development of the LBM non-isothermal flows with arbitrarily large Mach number." Вычислительные технологии, no. 1(26) (April 2, 2021): 62–71. http://dx.doi.org/10.25743/ict.2021.26.1.005.

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При решении задач динамики жидкостей и газов в области малых скоростей потока и при изотермических условиях с успехом применяется метод решеточных уравнений Больцмана (LBM). Для решения дискретного уравнения Больцмана может быть использован новый метод Particles-on-Demand (PonD), в котором в каждой точке сетки дискретизация функции распределения в пространстве скоростей центрирована относительно текущей скорости потока. В отличие от классического LBM, метод PonD применим не только для задач с малыми скоростями потока и при изотермических условиях. В данной работе реализован метод PonD D1Q5 с итерационным расчетом скорости переноса и явным расчетом первых трех моментов, включая скорости переноса. Показано, что рассмотренная модификация метода PonD хоть и накладывает ограничения на параметры, позволяет проводить расчеты в большем диапазоне допустимых скоростей. The purpose of the paper is to demonstrate applicability of the Particle on Demand (PonD) D1Q5 method with the explicit calculation of the first three moments to problem with high speed of the flow. The standard LBM is applicable for small flow velocities. Thus to overcome this limitation we use PonD. In this work, we use conservative version of PonD - the D1Q5 method with the explicit calculation of the first three moments. Methodology. The Pond over LBM was applied to the Riemann problem in order to demonstrate the advantage of the method. In this work, we choose the case when contact discontinuities could propagate at variable speed. Findings. If the interpolation pattern is fixed relative to the point at which there is a current update of the discrete distribution function, then the transfer step can be written explicitly, thus the scheme is conservative. On the other hand, this imposes additional restrictions on the temperature and the flow rate. But even if the PonD scheme is limited to a fixed interpolation pattern, it is possible to simulate flows with larger values of the Mach number than in the case when the classical method of lattice Boltzmann equations is used. Originality/value. In the described particular case of the PonD method, it is possible to avoid iterations by calculating the temperature and velocity values directly at a new time layer. In this work, we have investigated the properties and the range of applicability (admissible values of temperature and velocity) of such modification of PonD.

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GÖRSCH, D. "GENERALIZED DISCRETE VELOCITY MODELS." Mathematical Models and Methods in Applied Sciences 12, no. 01 (2002): 49–75. http://dx.doi.org/10.1142/s0218202502001544.

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Starting from a mesoscopic principle of moment conservation, discrete Boltzmann collision operators Jh are constructed, which both converge to bounded collision operators JΩ and have the same collision invariants as the original Boltzmann collision operator J. The crucial point of this construction is the application of a weak formulation of the gain operator to remove the post-collision velocities from it as well as the development of moment conserving integration formulas for the approximation of surface integrals over the unit sphere. Finally two applications for the discrete operators are presented.

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35

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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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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37

Ilyin, Oleg. "Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics." Mathematics 9, no. 9 (2021): 993. http://dx.doi.org/10.3390/math9090993.

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In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

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38

Baumann, G., and T. F. Nonnenmacher. "Bracket formulation for discrete two-velocity Boltzmann equations." Physics Letters A 122, no. 3-4 (1987): 149–52. http://dx.doi.org/10.1016/0375-9601(87)90794-8.

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Beale, J. Thomas. "Large-time behavior of discrete velocity boltzmann equations." Communications In Mathematical Physics 106, no. 4 (1986): 659–78. http://dx.doi.org/10.1007/bf01463401.

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Dubois, François, Tony Fevrier, and Benjamin Graille. "Lattice Boltzmann Schemes with Relative Velocities." Communications in Computational Physics 17, no. 4 (2015): 1088–112. http://dx.doi.org/10.4208/cicp.2014.m394.

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AbstractIn this contribution, a new class of lattice Boltzmann schemes is introduced and studied. These schemes are presented in a framework that generalizes the multiple relaxation times method of d’Humières. They extend also the Geier’s cascaded method. The relaxation phase takes place in a moving frame involving a set of moments depending on a given relative velocity field. We establish with the Taylor expansion method that the equivalent partial differential equations are identical to the ones obtained with the multiple relaxation times method up to the second order accuracy. The method is then performed to derive the equivalent equations up to third order accuracy.

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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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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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Feldman, Mikhail, and Seung-Yeal Ha. "Nonlinear Functionals of Multi-D Discrete Velocity Boltzmann Equations." Journal of Statistical Physics 114, no. 3/4 (2004): 1015–33. http://dx.doi.org/10.1023/b:joss.0000012515.85916.2a.

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Cornille, H. "Exact solutions for nonconservative two-velocity discrete Boltzmann models." Journal of Mathematical Physics 39, no. 4 (1998): 2004–18. http://dx.doi.org/10.1063/1.532274.

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Euler, Norbert, and Ove Lindblom. "On discrete velocity Boltzmann equations and the Painlevé analysis." Nonlinear Analysis: Theory, Methods & Applications 47, no. 2 (2001): 1407–12. http://dx.doi.org/10.1016/s0362-546x(01)00276-0.

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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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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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Cornille, Henri. "Two-velocity discrete boltzmann models: Positivity and theH-Theorem." Letters in Mathematical Physics 19, no. 3 (1990): 211–16. http://dx.doi.org/10.1007/bf01039314.

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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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Dubois, François, and Pierre Lallemand. "On Triangular Lattice Boltzmann Schemes for Scalar Problems." Communications in Computational Physics 13, no. 3 (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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