Hydrodynamic limits with shock waves of the Boltzmann equation

We show that piecewise smooth solutions with shocks of the Euler equations in gas dynamics can be obtained as the zero Knudsen number limit of solutions of the Boltzmann equation for hard sphere collision model. The construction of the Boltzmann solutions is done in two steps. First we introduce a generalized Hilbert expansion with shock layer correction to construct approximations to the solutions of the Boltzmann equations with small Knudsen numbers. We then apply the recently developed macro‐micro decomposition and energy method for Boltzmann shock layers to construct the exact Boltzmann solutions through the stability analysis. © 2004 Wiley Periodicals, Inc.     

Compressible Navier–Stokes approximation to the Boltzmann equation

Even though the system of the compressible Navier–Stokes equations is not a limiting system of the Boltzmann equation when the Knudsen number tends to zero, it is the second order approximation by applying the Chapman–Enskog expansion. The purpose of this paper is to justify this approximation rigorously in mathematics. That is, if the difference between the initial data for the compressible Navier–Stokes equations and the Boltzmann equation is of the second order of the Knudsen number, so is the difference between two solutions for all time. The analysis is based on a refined energy method for a fluid-type system using the techniques for the system of viscous conservation laws.     

Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.     

Numerical study of radiometric forces via the direct solution of the Boltzmann kinetic equation

The two-dimensional rarefied gas motion in a Crookes radiometer and the resulting radiometric forces are studied by numerically solving the Boltzmann kinetic equation. The collision integral is directly evaluated using a projection method, and second-order accurate TVD schemes are used to solve the advection equation. The radiometric forces are found as functions of the Knudsen number and the temperatures, and their spatial distribution is analyzed.     

Lattice Boltzmann simulation for high-speed compressible viscous flows with a boundary layer

In this study, the lattice Boltzmann method is employed for simulating high-speed compressible viscous flows with a boundary layer. The coupled double-distribution-function lattice Boltzmann method proposed by Li et al. (2007) is employed because of its good numerical stability and non-free-parameter feature. The non-uniform mesh construction near the wall boundary in fine grids is combined with an appropriate wall boundary treatment for the finite difference method in order to obtain accurate spatial resolution in the boundary layer problem. Three typical problems in high-speed viscous flows are solved in the lattice Boltzmann simulation, i.e., the compressible boundary layer problem, shock wave problem, and shock boundary layer interaction problem. In addition, in-depth comparisons are made with the non-oscillatory and non-free-parameter dissipation (NND) scheme and second order upwind scheme in the present lattice Boltzmann model. Our simulation results indicate the great potential of the lattice Boltzmann method for simulating high-speed compressible viscous flows with a boundary layer. Further research is needed (e.g., better numerical models and appropriate finite difference schemes) because the lattice Boltzmann method is still immature for high-speed compressible viscous flow applications.     

Application of lattice Boltzmann method for incompressible viscous flows

Because of the presence of corner eddies that change in number and pattern the lid-driven cavity problem has been found suitable to study various aspects of the performance of solution algorithms for incompressible viscous flows. It retains all the difficult flow physics and is characterized by a large primary eddy at the centre and secondary eddies located near the cavity corners. In this work, lid-driven cavity flow is simulated by lattice Boltzmann method with single-relaxation-time and it is compared with those by lattice Boltzmann method with multi-relaxation-time and finite difference method. The effects of the Reynolds number on the size, centre position and number of vortices are studied in detail together with the flow pattern in the cavity. The close agreement of the results bears testimony to the validity of this relatively new approach. However lattice Boltzmann method with multi-relaxation-time model is seen to remove the difficulties faces by the lattice Boltzmann method with single-relaxation-time at higher Reynolds numbers.     

Numerical simulation for the solitary wave of Zakharov–Kuznetsov equation based on lattice Boltzmann method

The Zakharov–Kuznetsov equation is considered, which is an equation describing two dimensional weakly nonlinear ion-acoustic waves in plasma. We focus on using the lattice Boltzmann method to study the Zakharov–Kuznetsov equation. A lattice Boltzmann model is constructed. In numerical experiments, the propagation of the single solitary wave and the collision of double solitary waves are simulated. The results with different parameters are investigated and compared.     

Heat transfer in a gas mixture between parallel plates

The one-dimensional steady-state heat flux and the temperature distribution in a rarefied gas mixture between two parallel plates with different temperatures are studied using the kinetic theory. The Boltzmann equation is solved by the projection method assuming that the gas consists of elastic hard spheres and the reflection from the surfaces is diffuse. The flow features are analyzed for a wide range of the Knudsen number. The molecular numerical densities of the components, the total temperature of the mixture, and the mixture heat flux are obtained. The behavior of the distribution functions for the components is discussed. A comparison with other authors’ results shows that the accuracy of the given method is good.     

Stability analysis of the plane Couette flow for a model kinetic equation

The stability of the plane Couette flow is studied using the simplified Boltzmann equation (the BGK equation) in which the high modes in the space of velocities and coordinates are truncated. The solution to the Navier-Stokes equation with small additional terms depending on the Knudsen number is used as the stationary solution. We assume that the perturbations depend only on the coordinate that is orthogonal to the flow. The density perturbations are assumed to be nonzero. In this approximation, the problem is found to be unstable in the case of small Knudsen numbers.     

Solution to the Boltzmann kinetic equation for high-speed flows

The Boltzmann kinetic equation is solved by a finite-difference method on a fixed coordinate-velocity grid. The projection method is applied that was developed previously by the author for evaluating the Boltzmann collision integral. The method ensures that the mass, momentum, and energy conservation laws are strictly satisfied and that the collision integral vanishes in thermodynamic equilibrium. The last property prevents the emergence of the numerical error when the collision integral of the principal part of the solution is evaluated outside Knudsen layers or shock waves, which considerably improves the accuracy and efficiency of the method. The differential part is approximated by a second-order accurate explicit conservative scheme. The resulting system of difference equations is solved by applying symmetric splitting into collision relaxation and free molecular flow. The steady-state solution is found by the relaxation method.     

A new lattice Boltzmann model for the laplace equation

In this paper, a new lattice Boltzmann equation which is independent of time is proposed. Based on the new lattice Boltzmann equation, some steady problems can be modeled by the lattice Boltzmann method. In the further study, the Laplace equation is investigated with the method of the higher-order moment of equilibrium distribution functions and a series of partial differential equations in different space scales. The numerical results show that the new method is effective.     

Lattice Boltzmann solver of Rossler equation

We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.     

Simulation of rarefied gas flows on the basis of the Boltzmann kinetic equation solved by applying a conservative projection method

Flows of a simple rarefied gas and gas mixtures are computed on the basis of the Boltzmann kinetic equation, which is solved by applying various versions of the conservative projection method, namely, a two-point method for a simple gas and gas mixtures with a small difference between the molecular masses and a multipoint method in the case of a large mass difference. Examples of steady and unsteady flows are computed in a wide range of Mach and Knudsen numbers.     

Lattice Boltzmann model for the perfect gas flows with near-vacuum region

It is known that the standard lattice Boltzmann method has near-vacuum limit, i. e., when the density is near zero, this method is invalid. In this letter, we propose a simple lattice Boltzmann model for one-dimensional flows. It possesses the ability of simulating near-vacuum area by setting a limitation of the relaxation factor. Thus, the model overcomes the disadvantage of non-physical pressure and the density. The numerical examples show these results are satisfactory.     

Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains

Combining an asymptotic analysis of the lattice Boltzmann method with a stability estimate, we are able to prove some convergence results which establish a strict relation to the incompressible Navier–Stokes equation. The proof applies to the lattice Boltzmann method in the case of periodic domains and for specific bounded domains if the Dirichlet boundary condition is realized with the bounce back rule.     

一类非线性偏微分方程的四阶格子Boltzmann模型

通过Chapman-Enskog展开技术和多尺度分析,建立了一种新的D1Q4带修正项的四阶格子Boltzmann模型,一类非线性偏微分方程从连续的Boltzmann方程得到正确恢复．统一了KdV和Burgers等已知方程类型的格子BGK模型,还首次给出了组合KdV-Burgers,广义Burgers—Huxley等方程...     

A lattice Boltzmann model for Maxwell’s equations

In this paper, a lattice Boltzmann model for the Maxwell’s equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called 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 third-order dispersion term and the fourth-order dissipation term. As results, the conservation law 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 conservation law equation.