On the stability structure for lattice Boltzmann schemes

The stability structure for lattice Boltzmann schemes has been introduced in Banda et al. (2006) [16], Junk and Yong (2007) [14] to analyze the stability of numerical algorithms. The first purpose of this paper is to discuss the stability structure from the perspective of matrix analysis. Its second goal is to illustrate and apply the results to different classes of lattice Boltzmann collision operators. In particular we formulate an equivalence condition–just recently also reported in Yong (2008) [18]–that guarantees the existence of a pre-stability structure. It is then illustrated by several examples, how this equivalence condition can be effectively employed for the systematic verification and construction of stable collision operators. Finally, we point out some shortcomings of the stability structure approach arising in certain cases.     

Quartic parameters for acoustic applications of lattice Boltzmann scheme

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are obtained in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach.     

On the small-scale dynamical behavior of lattice BGK and lattice Boltzmann schemes

In this paper, we report some comparative simulations between lattice BGK and lattice Bolzmann schemes for two-dimensional fluid flows. A quantitative assessment of the validity of the lattice BGK and lattice Bolzmann schemes is presented for the two-dimensional weakly compressibleKolmogorov flow. We use this flow to study the difference of the two schemes at small scales. A lowReynolds (R _e 300) number simulation shows the almost identical energy spectra for both schemes except for the small-scale dynamics of lattice Bolzmann which is more noisy. Because of the intrinsic difficulties of nonlinear stability analysis, we use numerical simulations to investigate which scheme is more stable. It turns out the lattice BGK is more stable. It turns out the lattice BGK is more robust than lattice Bolzmann by increasing theReynolds numbers. Detailed comparison with other methods (e.g., spectral method) remains to be done in the near future.     

An evaluation of lattice Boltzmann schemes for porous medium flow simulation

We quantitatively evaluate the capability and accuracy of the lattice Boltzmann equation (LBE) for modeling flow through porous media. In particular, we conduct a comparative study of the LBE models with the multiple-relaxation-time (MRT) and the Bhatnagar–Gross–Krook (BGK) single-relaxation-time (SRT) collision operators. We also investigate several fluid–solid boundary conditions including: (1) the standard bounce-back (SBB) scheme, (2) the linearly interpolated bounce-back (LIBB) scheme, (3) the quadratically interpolated bounce-back (QIBB) scheme, and (4) the multi-reflection (MR) scheme. Three-dimensional flow through two porous media—a body-centered cubic (BCC) array of spheres and a random-sized sphere-pack—are examined in this study. For flow past a BCC array of spheres, we validate the linear LBE model by comparing its results with the nonlinear LBE model. We investigate systematically the viscosity-dependence of the computed permeability, the discretization error, and effects due to the choice of relaxation parameters with the MRT and BGK schemes. Our results show unequivocally that the MRT–LBE model is superior to the BGK–LBE model, and interpolation significantly improves the accuracy of the fluid–solid boundary conditions.     

Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves

We follow the mathematical framework proposed by Bouchut (2003) [22] and present in this contribution a dual entropy approach for determining equilibrium states of a lattice Boltzmann scheme. This method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law. It appears as a good mathematical framework for establishing a “H-theorem” for the system of equations with discrete velocities. The dual entropy approach is used with D1Q3 lattice Boltzmann schemes for the Burgers equation. It conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition. Satisfactory numerical results for strong nonlinear shocks and rarefactions are presented. We prove also that the dual entropy approach can be applied with a D1Q3 lattice Boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics. We establish also a negative result: with the present framework, the dual entropy approach cannot be used for the shallow water equations.     

Lattice Boltzmann schemes for quantum applications

We review the basic ideas behind the quantum lattice Boltzmann equation (LBE), and present a few thoughts on the possible use of such an equation for simulating quantum many-body problems on both (parallel) electronic and quantum computers.     

Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.     

Optimal relaxation collisions for lattice Boltzmann methods

The optimal relaxation time of about 0.8090 has been proposed to balance the efficiency, stability, and accuracy at a given lattice size of numerical simulations with lattice Boltzmann methods. The optimal lattice size for a desired Reynolds number can be refined by reducing the Mach number for incompressible flows. The functioned polylogarithm polynomials are defined and used to express the lattice Boltzmann equations at different time scales and to analyze the impact of relaxation times and lattice sizes on truncation errors. Smaller truncation errors can be achieved when relaxation times are greater than 0.5 and less than 1.0. The steady-state lid-driven cavity flow was chosen to validate the code of lattice Boltzmann procedures. The applications of the optimal relaxation parameters numerically balance the stability, efficiency, and accuracy through Hartmann flow. The optimal relaxation time can also be used to select the initial lattice size for the channel flow over a square cylinder with a given Mach number.     

Adjoint lattice Boltzmann equation for parameter identification

The lattice Boltzmann equation is briefly introduced using moments to clearly separate the propagation and collision steps in the dynamics. In order to identify unknown parameters we introduce a cost function and adapt control theory to the lattice Boltzmann equation to get expressions for the derivatives of the cost function vs. parameters. This leads to an equivalent of the adjoint method with the definition of an adjoint lattice Boltzmann equation.To verify the general expressions for the derivatives, we consider two elementary situations: a linearized Poiseuille flow to show that the method can be used to optimize parameters, and a nonlinear situation in which a transverse shear wave is advected by a mean uniform flow. We indicate in the conclusion how the method can be used for more realistic situations.     

多重网格格子Boltzmann方法的并行算法

针对复杂流动数值模拟中的格子Boltzmann方法存在计算网格量大、收敛速度慢的缺点,提出了基于三维几何边界的多重笛卡儿网格并行生成算法,并基于该网格生成方法提出了多重网格并行格子Boltzmann方法（LBM）。该方法结合不同尺度网格间的耦合计算,有效减少了计算网格量,提高了收敛速度;而且测试结果也表明该并行算法具有良好的可扩展性。     

Optimizing lattice Boltzmann simulations for unsteady flows

We present detailed analysis of a lattice Boltzmann approach to model time-dependent Newtonian flows. The aim of this study is to find optimized simulation parameters for a desired accuracy with minimal computational time. Simulation parameters for fixed Reynolds and Womersley numbers are studied. We investigate influences from the Mach number and different boundary conditions on the accuracy and performance of the method and suggest ways to enhance the convergence behavior.     

Multi-relaxation-time lattice Boltzmann model for axisymmetric flows

In this paper, a lattice-Boltzmann equation (LBE) with multi relaxation times (MRT) is presented for axisymmetric flows. The model is an extension of a recent model with single-relaxation-time [Guo et al., Phys. Rev. E 79, 046708 (2009)], which was developed based on the axisymmetric Boltzmann equation. Due to the use of the MRT collision model, the present model can achieve better numerical stability. The model is validated by some numerical tests including the Hagen-Poiseuille flow, the pulsatile Womersley flow, and the external flow over a sphere. Numerical results are in excellent agreement with analytical solutions or other available data, and the improvement in numerical stability is also confirmed.     

Consistent initial conditions for lattice Boltzmann simulations

We propose a simple and effective iterative procedure to generate consistent initial conditions for the lattice Boltzmann equation (LBE) for incompressible flows with a given initial velocity field u₀. Using the Chapman-Enskog analysis we show that not only the proposed procedure effectively solves the Poisson equation for the pressure field p₀ corresponding to u₀, it also generates at the same time the initial values for the nonequilibrium distribution functions {f_α} in a consistent manner. This procedure is validated for the decaying Taylor–Green vortex flow in two dimensions and is shown to be particularly effective when using the generalized LBE with multiple relaxation times.     

MRT Lattice Boltzmann schemes for confined suspension flows

We introduce a novel multiple-relaxation time (modified MRT) Lattice Boltzmann scheme for simulation of confined suspension flow. Via careful tuning of the free eigenvalues of the collision operator we can substantially reduce the error in the so-called hydrodynamic radius. Its performance has been compared to that of the TRT scheme for several benchmark problems. We have found that the optimal value of the free eigenvalue depends on the curvature of the solid-fluid interfaces. Hence, we have investigated suspension flow problems, with confining boundaries of different curvatures. We have found that the modified MRT scheme is better suited for suspension flow in curved confining walls, while the TRT scheme is better for suspension flow confined between planar walls.With both schemes we have investigated problems for confined suspension flows, namely 1) drag forces experienced by spheres flowing in confining flow channels of different cross sections, and 2) the lubrication force between a sedimenting sphere and the end cap of a confining cylindrical capillary.     

Thermal boundary conditions for thermal lattice Boltzmann simulations

Consistent 2D and 3D thermal boundary conditions for thermal lattice Boltzmann simulations are proposed. The boundary unknown energy distribution functions are made functions of known energy distribution functions and correctors, where the correctors at the boundary nodes are obtained directly from the definition of internal energy density. This boundary condition can be easily implemented on the wall and corner boundary using the same formulation. The discrete macroscopic energy equation is also derived for a steady and fully developed channel flow to assess the effect of the boundary condition on the solutions, where the resulting second order accurate central difference equation predicts continuous energy distribution across the boundary, provided the boundary unknown energy distribution functions satisfy the macroscopic energy level. Four different local known energy distribution functions are experimented with to assess both this observation and the applicability of the present formulation, and are scrutinized by calculating the 2D thermal Poiseuille flow, thermal Couette flow, thermal Couette flow with wall injection, natural convection in a square cavity, and 3D thermal Poiseuille flow in a square duct. Numerical simulations indicate that the present formulation is second order accurate and the difference of adopting different local known energy distribution functions is, as expected, negligible, which are consistent with the results from the derived discrete macroscopic energy equation.     

An efficient swap algorithm for the lattice Boltzmann method

During the last decade, the lattice-Boltzmann method (LBM) as a valuable tool in computational fluid dynamics has been increasingly acknowledged. The widespread application of LBM is partly due to the simplicity of its coding. The most well-known algorithms for the implementation of the standard lattice-Boltzmann equation (LBE) are the two-lattice and two-step algorithms. However, implementations of the two-lattice or the two-step algorithm suffer from high memory consumption or poor computational performance, respectively. Ultimately, the computing resources available decide which of the two disadvantages is more critical. Here we introduce a new algorithm, called the swap algorithm, for the implementation of LBE. Simulation results demonstrate that implementations based on the swap algorithm can achieve high computational performance and have very low memory consumption. Furthermore, we show how the performance of its implementations can be further improved by code optimization.     

Entropic lattice Boltzmann method for simulation of binary mixtures

A new kinetic model for binary mixtures and its lattice Boltzmann (LB) discretization is formulated. In the hydrodynamic limit, the model recovers the Navier–Stokes and the Stefan–Maxwell binary diffusion equations, satisfies the indifferentiability principle, and is thermodynamically consistent. The present model is able to simulate mixtures with different Schmidt numbers and with a large molecular weight ratio of the components.     

General regularized boundary condition for multi-speed lattice Boltzmann models

The lattice Boltzmann method is nowadays a common tool for solving computational fluid dynamics problems. One of the difficulties of this numerical approach is the treatment of the boundaries, because of the lack of physical intuition for the behavior of the density distribution functions close to the walls. A massive effort has been made by the scientific community to find appropriate solutions for boundaries. In this paper we present a completely generic way of treating a Dirichlet boundary for two- and three-dimensional flat walls, edges or corners, for weakly compressible flows, applicable for any lattice topology. The proposed algorithm is shown to be second-order accurate and could also be extended for compressible and thermal flows.     

Comparison of different propagation steps for lattice Boltzmann methods

Several possibilities exist to implement the propagation step of lattice Boltzmann methods. This paper describes common implementations and compares the number of memory transfer operations they require per lattice node update. A performance model based on the memory bandwidth is then used to obtain an estimation of the maximum achievable performance on different machines. A subset of the discussed implementations of the propagation step are benchmarked on different Intel- and AMD-based compute nodes using the framework of an existing flow solver that is specially adapted to simulate flow in porous media, and the model is validated against the measurements. Advanced approaches for the propagation step like “A–A pattern” or “Esoteric Twist” require more programming effort but often sustain significantly better performance than non-naïve but straightforward implementations.