A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

Robust treatment of no-slip boundary condition and velocity updating for the lattice-Boltzmann simulation of particulate flows

In the past decade, the lattice-Boltzmann method (LBM) has emerged as a very useful tool in studies for the direct-numerical simulation of particulate flows. The accuracy and robustness of the LBM have been demonstrated by many researchers; however, there are several numerical problems that have not been completely resolved. One of these is the treatment of the no-slip boundary condition on the particle-fluid interface and another is the updating scheme for the particle velocity. The most common used treatment for the solid boundaries largely employs the so-called “bounce-back” method (BBM). [Ladd AJC. Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part I. Theoretical foundation. J Fluid Mech (1994);271:285; Ladd AJC. Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part II. Numerical results. J Fluid Mech (1994);271:311.] This often causes distortions and fluctuations of the particle shape from one time step to another. The immersed boundary method (IBM), which assigns and follows a series of points in the solid region, may be used to ensure the uniformity of particle shapes throughout the computations. To ensure that the IBM points move with the solid particles, a force density function is applied to these points. The simplest way to calculate the force density function is to use a direct-forcing scheme. In this paper, we conduct a complete study on issues related to this scheme and examine the following parameters: the generation of the forcing points; the choice of the number of forcing points and sensitivity of this choice to simulation results; and, the advantages and disadvantages associated with the IBM over the BBM. It was also observed that the commonly used velocity updating schemes cause instabilities when the densities of the fluid and the particles are close. In this paper, we present a simple and very effective velocity updating scheme that does not only facilitate the numerical solutions when the particle to fluid density ratios are close to one, but also works well for particle that are lighter than the fluid.     

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

An upwind discretization scheme for the finite volume lattice Boltzmann method

The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys 1992;68:401–7; Peng G, Xi H, Duncan C, Chou SH. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82; Chen H. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys Rev E 1998;58:3955–63] or Finite Element [Lee T, Lin CL. A characteristic Galerkin method for discrete Boltzmann equation. J Comp Phys 2001;171:336–56; Shi X, Lin J, Yu Z. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 2003;42:1249–61] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods.We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.     

Lattice Boltzmann Models for Anisotropic Diffusion of Images

The lattice Boltzmann method has attracted more and more attention as an alternative numerical scheme to traditional numerical methods for solving partial differential equations and modeling physical systems. The idea of the lattice Boltzmann method is to construct a simplified discrete microscopic dynamics to simulate the macroscopic model described by the partial differential equations. The use of the lattice Boltzmann method has allowed the study of a broad class of systems that would have been difficult by other means. The advantage of the lattice Boltzmann method is that it provides easily implemented fully parallel algorithms and the capability of handling complicated boundaries. In this paper, we present two lattice Boltzmann models for nonlinear anisotropic diffusion of images. We show that image feature selective diffusion (smoothing) can be achieved by making the relaxation parameter in the lattice Boltzmann equation be image feature and direction dependent. The models naturally lead to the numerical algorithms that are easy to implement. Experimental results on both synthetic and real images are described.     

The Wigner–Boltzmann Monte Carlo method applied to electron transport in the presence of a single dopant

The Wigner–Boltzmann model is a partial integro-differential equation which describes the time dependent dynamics of quantum mechanical phenomena including the effects of lattice vibration as a second-order approximation. Recently a Monte Carlo technique exploiting the concept of signed particles has been developed for its ballistic counterpart, in one and two-dimensional space. In this work, we introduce an extension to the Wigner–Boltzmann model in three-dimensional geometries adapted for the treatment of the scattering term. As an application, we study the dynamics of an electron wave packet in proximity of a Coulombic potential in the presence of absorbing boundary conditions. This mimics the presence of a dopant atom buried in a semiconductor substrate. By using this method, one can observe how the lattice temperature eventually affects the dynamics of the wave packet.     

A lattice Boltzmann approach for the non-Newtonian effect in the blood flow

A new lattice Boltzmann approach within the framework of D2Q9 lattice for simulating shear-thinning non-Newtonian blood flows described by the power-law, Carreau-Yasuda and Casson rheology models is proposed in this study. The essence of this method lies in splitting the complete non-Newtonian effect up into two portions: one as the Newtonian result and the other as an effective external source. This arrangement takes the advantage in remaining fixed relaxation time during the whole course of numerical simulation that can avoid the potential numerical instability caused by the relaxation time approaches to 1/2, an inherent difficulty in the conventional lattice Boltzmann methods using varying relaxation times for the non-Newtonian effect. Macroscopically, consistency of the proposed model with the equations of motion for the three target non-Newtonian models is demonstrated through the technique of Chapman-Enskog multi-scale expansion. The feasibility and accuracy of the method are examined by comparing with the analytical solutions of the two-dimensional Poiseuille flows based on the power-law and Casson models. The results show that the velocity profiles agree very well with those of analytical solutions and the error analyses demonstrate that the proposed scheme is with second-order accuracy. The present approach also demonstrates its superiority over the conventional lattice Boltzmann method in the extent of numerical stability for simulating the power-law-based shear-thinning flows. The straightforwardness in scheme derivation and implementation renders the present approach as a potential method for the complex non-Newtonian flows.     

Application of lattice Boltzmann method in free surface flow simulation of micro injection molding

The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.     

Lattice Boltzmann computations of incompressible laminar flow and heat transfer in a constricted channel

A multi-population thermal lattice Boltzmann method (TLBM) is applied to simulate incompressible steady flow and heat transfer in a two-dimensional constricted channel. The method is validated for velocity and temperature profiles by comparing with a finite element method based commercial solver. The results indicate that, at various Reynolds numbers, the average flow resistance increases and the heat transfer rate decreases in a constricted channel in comparison to a straight channel. The effect of the constriction ratio is also investigated. The results show that the presented numerical model is a promising tool in analyzing simultaneous solution of fluid flow and heat transfer phenomena in complex geometries.     

A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows

A lattice Boltzmann model for simulating isothermal micro flows has been proposed by us recently [Niu XD, Chew YT, Shu C. A lattice Boltzmann BGK model for simulation of micro flows. Europhys Lett 2004;67(4):600]. In this paper, we extend the model to simulate the micro thermal flows. In particular, the thermal lattice Boltzmann equation (TLBE) [He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmann method in incompressible limit. J Comput Phys 1998;146:282] is used with modification of the relaxation times linking to the Knudsen number. The diffuse scattering boundary condition (DSBC) derived in our early model is extended to consider temperature jump at wall boundaries. Simple theoretical analyses of the DSBC are presented and the results are found to be consistent with the conventional velocity slip and temperature jump boundary conditions. Numerical validations are carried out by simulating two-dimensional thermal Couette flows and developing thermal flows in a microchannel, and the obtained results are found to be in good agreement with those given from the direct simulation Monte Carlo (DSMC), the molecular dynamics (MD) approaches and the Maxwell theoretical prediction.     

A cache‐efficient implementation of the lattice Boltzmann method for the two‐dimensional diffusion equation

The lattice Boltzmann method is an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. However, to achieve the scalability and speed potential of the lattice Boltzmann technique, the issues of data reusability in cache‐based computer architectures must be addressed. Utilizing the two‐dimensional diffusion equation, , this paper examines cache optimization for the lattice Boltzmann method in both serial and parallel implementations. In this study, speedups due to cache optimization were found to be 1.9–2.5 for the serial implementation and 3.6–3.8 for the parallel case in which the domain decomposition was optimized for stride‐one access. In the parallel non‐cached implementation, the method of domain decomposition (horizontal or vertical) used for parallelization did not significantly affect the compute time. In contrast, the cache‐based implementation of the lattice Boltzmann method was significantly faster when the domain decomposition was optimized for stride‐one access. Additionally, the cache‐optimized lattice Boltzmann method in which the domain decomposition was optimized for stride‐one access displayed superlinear scalability on all problem sizes as the number of processors was increased. Copyright © 2004 John Wiley & Sons, Ltd.     

Some results on energy-conserving lattice Boltzmann models

We consider the problem of “energy conserving” lattice Boltzmann models. A major difficulty observed in previous studies is the coupling between the viscous and thermal waves even at moderate wave numbers. We propose a theoretical framework based on the knowledge of the partial equivalent equations of the lattice Boltzmann scheme at several orders of precision. With the help of linearized models (inviscid and dissipative advective acoustics and classical acoustics), we suggest natural sets of relations for the parameters of lattice Boltzmann schemes. The application is proposed for three two-dimensional schemes. Numerical test cases for simple linear and nonlinear waves establish that the main difficulty in the previous contributions can now be overcome.     

A fractional step lattice Boltzmann method for simulating high Reynolds number flows

A fractional step lattice Boltzmann scheme is presented to greatly improve the stability of the lattice Boltzmann method (LBM) in modelling incompressible flows at high Reynolds number. This method combines the good features of the conventional LBM and the fractional step technique. Through the fractional step, the flow at an extreme case of infinite Reynolds number (inviscid flow) can be effectively simulated. In addition, the non-slip boundary condition can be directly implemented.     

Numerical Studies Based on Higher-Order Accuracy Lattice Boltzmann Model for the Complex Ginzburg-Landau Equation

In this paper, a higher-order accuracy lattice Boltzmann model for the complex Ginzburg-Landau equation is proposed. In order to obtain higher-order accuracy of truncation error and to overcome the drawbacks of ??error rebound?? in the previous models, a new assumption of additional distribution is presented to improve the accuracy of the model for the complex partial differential equation with nonlinear source term. As results, the complex Ginzburg-Landau equation is recovered with the fourth-order accuracy of truncation error. Based on this model, the problems of a single spiral wave in two-dimensional (2D) space and a single scroll in three-dimensional (3D) space are implemented to test the lattice Boltzmann scheme. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex Ginzburg-Landau equation.     

Progress of current seamless virtual boundary method

In this paper, a current virtual boundary method, i.e., a seamless virtual boundary method (VBM) is presented. In the seamless VBM, the forcing term is added not only to the grid points near the boundary but also to the grid points inside the boundary, in order to remove unphysical oscillations near the boundary. The development of seamless VBM can be applied to solve for heat transfer and moving boundary problems in both the Cartesian and the curvilinear coordinates, and the lattice Boltzmann equation. A series of the method is validated in the typical test problems. Therefore, it is concluded that the present method is a very versatile numerical approach for solving the incompressible Navier–Stokes equations.     

Lattice Boltzmann based PDE solver on the GPU

In this paper, we propose a hardware-accelerated PDE (partial differential equation) solver based on the lattice Boltzmann model (LBM). The LBM is initially designed to solve fluid dynamics by constructing simplified microscopic kinetic models. As an explicit numerical scheme with only local operations, it has the advantage of being easy to implement and especially suitable for graphics hardware (GPU) acceleration. Beyond the Navier–Stokes equation of fluid mechanics, a typical LBM can be modified to solve the parabolic diffusion equation, which is further used to solve the elliptic Laplace and Poisson equations with a diffusion process. These PDEs are widely used in modeling and manipulating images, surfaces and volumetric data sets. Therefore, the LBM scheme can be used as an GPU-based numerical solver to provide a fast and convenient alternative to traditional implicit iterative solvers. We apply this method to several examples in volume smoothing, surface fairing and image editing, achieving outstanding performance on contemporary graphics hardware. It has the great potential to be used as a general GPU computing framework for efficiently solving PDEs in image processing, computer graphics and visualization.     

Meshless kinetic upwind method for compressible, viscous rotating flows

This paper presents the split-stencil least square kinetic upwind method for Navier–Stokes (SLKNS) solver using kinetic flux vector splitting (KFVS) scheme with Chapman-Enskog distribution. SLKNS solver operates on an arbitrary distribution of points and uses a novel least squares method which differs from the normal equations approach as it generates the non-symmetric cross-product matrix by selective splitting of the set of neighbours to avoid ill-conditioning. SLKNS also uses the axi-symmetric formulation of the Boltzmann equation and kinetic slip boundary condition. SLKNS is capable of capturing weak secondary flows as well as features of strong rotation characterized by steep density gradient and thin boundary layers towards the peripheral region with a rarefied central core.     

Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids

We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution.     

Numerical Method Based on the Lattice Boltzmann Model for the Kuramoto-Sivashinsky Equation

In this paper, we proposed a lattice Boltzmann model based on the higher-order moment method for the Kuramoto-Sivashinsky equation. A series of partial differential equations obtained by using multi-scale technique and Chapman-Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the fifth-order dispersion term and the sixth-order dissipation term. As results, the Kuramoto-Sivashinsky equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the Kuramoto-Sivashinsky equation.