Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom

A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended to compute complex-geometry three-dimensional flows of a diatomic gas with rotational degrees of freedom.     

A lattice Boltzmann model for the Fokker-Planck equation

In this paper, a lattice Boltzmann model is presented for solving one and two-dimensional Fokker-Planck equations with variable coefficients. In particular, it is efficient to simulate one-dimensional stochastic processes governed by the Fokker-Planck equation. Numerical results agree well with the exact solutions, which indicates that the proposed model is suitable for solving the Fokker-Planck equation.     

Kinetic model of the Boltzmann equation with limiting gas flow regimes at low Knudsen numbers

A new model of the Boltzmann kinetic equation is constructed that describes both slow nonisothermal and Navier-Stokes continuum gas flows. The model is used to compute the slow nonisothermal flow past a circular cylinder. It is shown that the force exerted by the gas on the cylinder is affected by thermal stresses.     

A novel lattice Boltzmann model for the Poisson equation

In this paper, a novel lattice Boltzmann model is proposed to solve the Poisson equation through modifying equilibrium distribution function. Compared with previous models, which can be viewed as the solvers to diffusion equation, the present model is a genuine solver to the Poisson equation, and the transient term derived by previous models is eliminated. Numerical solutions agree well with analytical solutions, which indicates the potential of the present model for solving the Poisson equation.     

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.     

Fluid-dynamic limit for the Ruijgrok-Wu model of the Boltzmann equation

In this paper we consider the fluid-dynamic limit for the Ruijgrok-Wu model derived from the Boltzmann equation. We use new technique developed in [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254] in order to get the convergence. First, we obtain the approximate transport equation for the given kinetic model. Then using the averaging lemma, we obtain the convergence. This paper shows how to relate the given kinetic model with the averaging lemma to get the convergence.     

On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas

The Broadwell model of the Boltzmann equation for a simple discrete velocity gas is investigated on two asymptotic problems. (a) The decay of solutions inxR ast+. (b) The hydrodynamical limit in the compressible Euler level as the mean free path0.     

Lattice Boltzmann model for two-dimensional unsteady Burgers’ equation

In this paper, a special lattice Boltzmann model is proposed to simulate two-dimensional unsteady Burgers’ equation. The maximum principle and the stability are proved. The model has been verified by several test examples. Excellent agreement is obtained between numerical predictions and exact solutions. The cases of steep oblique shock waves are solved and compared with the two-point compact scheme results. The study indicates that lattice Boltzmann model is highly stable and efficient even for the problems with severe gradients.     

Lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients

In this paper, lattice Boltzmann model for a generalized Gardner equation with time-dependent variable coefficients, which can provide some more realistic models than their constant-coefficient counterparts, is derived through selecting equilibrium distribution function and adding the compensate function, appropriately. Effects and approximate value range of the free parameters, which are introduced to adjust the single relaxation time and equilibrium distribution function, are discussed in detail, as well as the impact of the lattice space step and velocity. Numerical simulations in different situations of this equation are conducted, including the propagation and interaction of the solitons, the evolution of the non-propagating soliton and the propagation of the double-pole solutions. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current lattice Boltzmann model is a satisfactory and efficient algorithm.     

On a simulation scheme for the Boltzmann equation

A scheme for the simulation of solutions of the Boltzmann equation derived by Nanbu is investigated. Rigorous results concerning questions of justification, the computation effort and the energy fluctuations are presented.     

Laguerre polynomial solution of the nonlinear Boltzmann equation for the hard-sphere model

We consider a spatially homogeneous and isotropic gas consisting of hard-sphere molecules. A vector representation of the scattering kernel is used to adapt the original Boltzmann equation to the idealized geometrical situation. By means of an expansion of the distribution function in terms of Laguerre polynomials this scalar Boltzmann equation is transformed to a set of moment equations. All algebraized collision integrals can be evaluated analytically. We discuss the truncation of the moment equations necessary for the practical application of this method. The eigenvalues of the linearized relaxation problem show a good convergence with respect to the truncation index.

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Zusammenfassung Wir betrachten ein räumlich homogenes Gas harter Kugeln mit isotroper Geschwindigkeitsverteilung. Mit Hilfe einer Vektordarstellung des Streukerns wird die nichtlineare Boltzmanngleichung den vereinfachten geometrischen Verhältnissen angepaßt. Die entstehende skalare kinetische Gleichung wird durch eine Laguerre-Reihenentwicklung der Teilchenverteilungsdichte in ein System von Momentegleichungen übergeführt. Sämtliche algebraisierten Stoßintegrale erweisen sich als analytisch lösbar. Wir diskutieren den für den praktischen Gebrauch der Methode notwendigen Abbruch des Systems der Momentegleichungen. Die Eigenwerte des linearisierten Relaxationsproblems zeigen eine rasche Konvergenz bezüglich einer Steigerung des Abbruchindex.

     

Particle simulations of the Kac model of the Boltzmann equation

We consider a simplified mathematical model of the Boltzmann equation which was introduced by Kac. A quasi-Monte-Carlo particle simulation using (t, m, s)-nets is described. The particle movement is shown to be an evaluation of the volume of a subset of I⁴=[0,1)⁴. An error bound for quasi-Monte-Carlo approximation of the volume of a generalized quadrant set is derived, when a (t, m, s)-net is used. Convergence of the simulation is proved when the particles are reordered according to their velocity in every time step. Quasi-Monte-Carlo and Monte-Carlo particle simulations are compared in computational experiments. The results indicate that quasi-Monte-Carlo simulation outperforms standard Monte-Carlo approaches.     

Scaling limits for the Ruijgrok–Wu model of the Boltzmann equation

The Ruijgrok–Wu model of the kinetic theory of rarefied gases is investigated both in the fluid‐dynamic and hydro‐dynamic scalings. It is shown that the first limit equation is a first order quasilinear conservation law, whereas the limit equation in the diffusive scaling is the viscous Burgers equation. The main difficulties came from initial layers that we handle here. Copyright © 2001 John Wiley & Sons, Ltd.     

Second Darboux problem for the wave equation with a power-law nonlinearity

For the one-dimensional wave equation with a power-law nonlinearity, we consider the second Darboux problem and study the existence and uniqueness of a global solution, the existence of local solutions, and the absence of global solutions.     

Convergence of a splitting method for the nonlinear Boltzmann equation

Convergence of a splitting method scheme for the nonlinear Boltzmann equation is considered. Using the splitting method scheme, boundedness of the positive solutions in a space of continuous functions is obtained. By means of the solution boundedness and some a priori estimates, convergence of the splitting method scheme and uniqueness of the limiting element are proved. The limiting element satisfies an equivalent integral Boltzmann equation. Thereby global in time solvability of the nonlinear Boltzmann equation is shown.     

Some solvability problems for the Boltzmann equation in the framework of the Shakhov model

We consider the nonlinear Boltzmann equation in the framework of the Shakhov model for the classical problem of gas flow in a plane layer. The problem reduces to a system of nonlinear integral equations. The nonlinearity of the studied system can be partially simplified by passing to a new argument depending on the solution of the problem itself. We prove the existence theorem for a unique solution of the linear system and the existence theorem for a positive solution of the nonlinear Urysohn equation. We determine the temperature jumps on the lower and upper walls in the linear and nonlinear cases, and it turns out that the difference between them is rather small.     

Uniform stability of solutions of Boltzmann equation for soft potential with external force

The study of Cauchy problem of the Boltzmann equation is important in both theory and applications. Existence of global solutions to the equation and uniform stability of solutions in the absence of external force were introduced in the previous work on the Boltzmann equation. In this paper, we will investigate the uniform stability of solutions in L¹ for the Cauchy problem of the Boltzmann equation when there is an external force for the case of soft potentials.