Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation

In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL -norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.     

Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Analytic solution of the ellipsoidal-statistical model Boltzmann equation

An exact solution of the ellipsoidal-statistical model Boltzmann equation is constructed. The problem of the temperature jump in a rarefied gas is considered by way of illustration. By expanding the distribution function in two orthogonal directions the problem is reduced to the solution of a vector transport equation with polynomial boundary conditions. The Case approach reduces the equation to a characteristic equation for which generalized eigenvectors and eigenvalues are found. A theorem of existence and uniqueness of the solution, represented in the form of an expansion in eigenvectors, is proved. The proof reduces to solving a Riemann-Hilbert vector boundary-value problem with a matrix coefficient whose diagonalizing matrix has branch points in the complex plane. The value of the temperature jump is found from the conditions of solvability of the boundary-value problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 151–164, March–April, 1992.     

An asymptotic method for a singular hyperbolic equation

Ho & Meyer [1] have sketched an argument for arriving at asymptotic properties of solutions of a pair of non-linear conservation laws with floating, shocktype boundary condition of importance in gas dynamics and oceanography. To clarify the mathematical basis of their method of inequalities, a proof of their main asymptotic results is given here. It depends on substantially weakened hypotheses.     

Integral form of the Boltzmann equation for the forced diffusion of charged particles in anisotropically scattering media

Summary According to a recently proposed integral formulation of the Boltzmann equation for test particles in an external force field, the present paper shows how it is actually possible to reduce the original transport problem to the solution of a system of linear integral equations in one variable for a wide and physically meaningful class of interaction laws between test particles and host medium. The feasibility study deals mainly with anisotropic scattering, and provides the kernels of the final integral system to be solved. The explicit analytic form of such kernels has been explicitly worked out for some interaction models of particular interest.

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Sommario Sulla base di una recente formulazione della teoria del trasporto di particelle in presenza di una forza esterna, il presente lavoro mostra come il problema considerato sia riconducibile, per varie e significative leggi di interazione tra le particelle e il mezzo in cui diffondono, a un sistema di equazioni integrali lineari in una variabile indipendente. Viene studiato soprattutto lo scattering anisotropo, e sono ricavati in forma esplicita i kernel del sistema integrale per alcuni modelli di interazione di maggiore interesse.

     

Some applications of the method of moments for the homogeneous Boltzmann and Kac equations

Using the method of moments, we prove that any polynomial moment of the solution of the homogeneous Boltzmann equation with hard potentials or hard spheres is bounded provided that a moment of order strictly higher than 2 exists initially. We also give partial results of convergence towards the Maxwellian equilibrium in the case of soft potentials. Finally, exponential as well as Maxwellian estimates are introduced for the Kac equation.     

污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。     

A lattice Boltzmann model for diffusion of binary gas mixtures that includes diffusion slip

This paper describes the development of a lattice Boltzmann (LB) model for a binary gas mixture, and applications to channel flow driven by a density gradient with diffusion slip occurring at the wall. LB methods for single component gases typically use a non‐physical equation of state in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi‐component systems containing gases of differing molecular masses. Substantial variations in the species densities and pressures may exist even at low Mach numbers; hence, the usual linearized equation of state for small fluctuations is unsuitable. Also, existing methods for implementing boundary conditions do not extend easily to novel boundary conditions, such as diffusion slip. The new model developed for multi‐component gases avoids the pitfalls of some other LB models. A single computational grid is shared by all the species, and the diffusivity is independent of the viscosity. The Navier–Stokes equation for the mixture and the Stefan–Maxwell diffusion equation are both recovered by the model. Diffusion slip, the non‐zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one‐dimensional model for channel flow. To increase the accuracy of the scheme, a second‐order numerical implementation is needed. This may be achieved using a variable transformation method that does not increase the computational time. Simulations were carried out on hydrogen and water diffusion through a narrow channel for varying total pressure and concentration gradients. Copyright © 2011 John Wiley & Sons, Ltd.     

Study of a new semi-continuous model of the Boltzmann equation

For a semi-continuous model of the Boltzmann equation (1) peculiar solutions are obtained and generally the global existence of solutions of the initial value problem is discussed. The global existence is possible even in some cases for partially negative initial number densities, which are not physical problems, but mathematical ones. It can be shown that in some cases the entropy begins to increase, reaches a maximum and decreases again.     

Insights on a sign‐preserving numerical method for the advection–diffusion equation

In this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A higher‐order accuracy lattice Boltzmann model for the wave equation

A lattice Boltzmann model with higher‐order accuracy for the wave motion is proposed. The new model is based on the technique of the higher‐order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The forms of moments are derived from the binary wave equation by designing the higher‐order dissipation and dispersion terms. The numerical results agree well with classical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

The asymptotic expression of the solution of the Cauchy's problem for a higher order linear ordinary differential equation when the limit equation has singularity

In this paper we consider the asymptotic expression of the solution of the Cauchy's problem for a higher order equation when the limit equation has singularity. In order to construct the asymptotic expression of the solution, the region is divided into three sub-areas. In every small region, the solution of the differential equation is different. Project supported by the National Natural Science Foundation of China     

Exponential spline interpolation in characteristic based scheme for solving the advective–diffusion equation

This paper demonstrates the use of shape‐preserving exponential spline interpolation in a characteristic based numerical scheme for the solution of the linear advective–diffusion equation. The results from this scheme are compared with results from a number of numerical schemes in current use using test problems in one and two dimensions. These test cases are used to assess the merits of using shape‐preserving interpolation in a characteristic based scheme. The evaluation of the schemes is based on accuracy, efficiency, and complexity. The use of the shape‐preserving interpolation in a characteristic based scheme is accurate, captures discontinuities, does not introduce spurious oscillations, and preserves the monotonicity and positivity properties of the exact solution. However, fitting exponential spline interpolants to the nodal concentrations is computationally expensive. Exponential spline interpolants were also fitted to the integral of the concentration profile. The integral of the concentration profile is a smoother function than the concentration profile. It requires less computational effort to fit an exponential spline interpolant to the integral than the nodal concentrations. By differentiating the interpolant, the nodal concentrations are obtained. This results in a more efficient and more accurate numerical scheme. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

The cumulant method for the space-homogeneous Boltzmann equation

In this work we give a comparison of the exact Bobylev/Krook-Wu solution to the space-homogeneous Boltzmann equation and numerical results obtained by a implementation of the cumulant method for the space-homogeneous case. We find excellent agreement of the numerical solution to the cumulant equations with the exact solution of the space-homogeneous Boltzmann equation as long as the exact, non-linear production terms are used. If a linearized variant of the production terms is used, relaxation rates may be underestimated due to convergence to the solution of the linearized equations.Received: 3 April 2004, Accepted: 3 September 2004, Published online: 22 February 2005PACS: 51.10. + y, 51.30. + i, 47.11 + j, 47.45.-n Correspondence to: K.H. Hoffmann     

Global Lp-properties for the spatially homogeneous Boltzmann equation

This paper studies the L ^p-behavior for 1p of solutions of the nonlinear, spatially homogeneous Boltzmann equation for a class of collision kernels including inverse k ^th-power forces with k>5 and angular cut-off. The following topics are treated: differentiability in L ^p together with global boundedness in time for L ^p-moments that exist initially, translation continuity in L ^p uniformly in time, and strong convergence to equilibrium.