Nonlinear Collision Integral Research Articles

Finite difference method for numerical solution of the initial and boundary value problem for boltzmann’s sixmoment system of equations

Boltzmann’s one-dimensional non-linear non-stationary moment system of equations in the third approximation is presented, in which the first, third and fourth equations corresponds to the laws of conservation of mass, momentum and energy, respectively. This system contains six equations and represents a nonlinear system of hyperbolic type equations. For the Boltzmann’s six- moment system of equations an initial and boundary value problem is formulated. The macroscopic boundary condition contains the moments of the incident particles distribution function on the boundary and moments of the reflected particles distribution function from the boundary. The boundary condition depends on the temperature of the wall (boundary). In this work, using the finite-difference method, an approximate solution of the mixed problem for the Boltzmann system of moment equations is constructed in the third approximation under the boundary conditions obtained by approximating the Maxwell boundary condition. For given values of the coefficients included in the moments of the nonlinear collision integral and the parameter depending on the wall temperature, as well as for fixed values of the initial conditions, a numerical experiment was carried out. As a result, the approximate values of the particle distribution function incident on the boundary and reflected from the boundary, as well as the density, temperature and average velocity of gas particles, as moments of the particle distribution function, are obtained.

A New Stability and Convergence Proof of the Fourier--Galerkin Spectral Method for the Spatially Homogeneous Boltzmann Equation

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier--Galerkin spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217--1245] has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Despite its practical success, the stability of the method was only recently proved in [F. Filbet and C. Mouhot, Trans. Amer. Math. Soc., 363 (2011), pp. 1947--1980] by utilizing the “spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.

Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

Abstract In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.

Metriplectic integrators for the Landau collision operator

We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations, in general, independently of the mesh configuration.

Conservative discretization of the Landau collision integral

We describe a density-, momentum-, and energy-conserving discretization of the nonlinear Landau collision integral. The method is suitable for both the finite-element and discontinuous Galerkin methods and does not require structured meshes. The conservation laws for the discretization are proven algebraically and demonstrated numerically for an axially symmetric nonlinear relaxation problem using a finite-element implementation.

Recurrence procedure for calculating kernels of the nonlinear collision integral of the Boltzmann equation

A recurrence procedure for a sequential construction of kernels \(G_{{l_1},{l_2}}^l\) (c, c1, c2) appearing upon the expansion of a nonlinear collision integral of the Boltzmann equation in spherical harmonics is developed. The starting kernel for this procedure is kernel G0,00 (c, c1, c2) of the collision integral for the distribution function isotropic with respect to the velocities. Using the recurrence procedure, a set of kernels \(G_{{l_1},{l_2}}^{ + l}\) (c, c1, c2) for a gas consisting of hard spheres and Maxwellian molecules is constructed. It is shown that the resultant kernels exhibit similarity and symmetry properties and satisfy the relations following from the conservation laws.

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels Ll(v, v1) of linear collision integral and kernels Gl, 0l(v, v1, v2) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels Gl, 0+l(v, v1, v2) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.

Construction of kernels of the nonlinear collision integral in the Boltzmann equation using laplace transformation

It is shown that the relation between kernels L l (v, v 1) of the linear collision integral and kernels G ,0 (v, v 1, v 2) of the nonlinear collision integral can be reduced to the Laplace transformation. Analytic expressions for nonlinear kernels G 0,0 +0 (v, v 1, v 2) and G 1,0 +1 (v, v 1, v 2) are determined for hard spheres and pseudo-Maxwellian molecules.

Computer simulation and analysis of the Holweck pump in the transient regime

The results of simulating the Holweck pump by numerically solving the Boltzmann kinetic equation are reported. The nonlinear collision integral is calculated using the conservative projection method. The translation operator is approximated with tetrahedral grids. The ratio of pressures in pumped-in and pumpedout containers is studied as a function of the gas density (rarefaction) and the radius and rotation velocity of the rotor at Knudsen numbers close to unity.

Some general properties of the nonlinear collision integral in the Boltzmann equation

A new method for computing matrix elements of the collision integral in the Boltzmann equation makes it possible to consider many problems of the kinetic theory of gases in a new way. Nonlinear kernels of the collision integral are studied and similarity relations, which simplify significantly the problem of constructing of such kernels, are proved.

On the model kinetic description of plasma and a Boltzmann gas of hard spheres

A new form of the collision operator for a Boltzmann gas of hard spheres and Coulombplasma is proposed. One-component and many-component systems are considered. Theproposed collision operator properly takes into account the relaxation of the first 13hydrodynamic moments. Besides this, it accounts for the non-diagonal componentcontribution in the quadratic approximation in the expansion of the linearized collisionoperator with respect to the complete system of Hermite polynomials. It is shown thatfor a system of charged particles with the Coulomb interaction potential, thesecontributions are essential and lead to Spitzer corrections to the transport coefficients. Anexpression for the intensity of the Langevin source in the kinetic equation is obtainedin the same approximation. A new form of the model collision operator for aBoltzmann gas of hard spheres is proposed. For a many-component system we havereconstructed a non-linear model collision integral by using the linearized collisionintegral found. Unlike previous ones, it does not contain complicated exponentialdependence and avoids coefficient ambiguity in the many-component collision integral.

An approach to entropy consistency in second-order hydrodynamic equations

As the flow becomes rarefied it has been seen that predictions of continuum formulations, such as the Navier–Stokes equations, become inaccurate. These inaccuracies stem from the linear approximations to the stress and heat flux in the viscous flux terms in the Navier–Stokes equations. Hence, it has long been conjectured that the inclusion of higher-order terms in the constitutive relations for the stress and heat flux may improve the predictive capabilities of such continuum formulations. Following this approach, second-order systems of hydrodynamic equations, such as the Burnett and Woods equations, were applied to the shock structure problem. While it was observed that these equations afforded a better description of the shock structure on coarse grids, they were prone to small wavelength instabilities when the grids were refined. The cause of this instability was subsequently traced to the fact that these equations can potentially violate the second law of thermodynamics when the local Knudsen number exceeds a critical limit. This leads to the fundamental question: is entropy consistency achievable in a system of second-order hydrodynamic equations? To answer this question, a novel set of equations, known as the BGK-Burnett equations, is constructed by taking moments of the Boltzmann equation for the second-order distribution function.The formulation of second-order hydrodynamic equations by moment methods is beset by three hurdles: (i) the highly nonlinear collision integral in the Boltzmann equation needs to be evaluated, (ii) the second-order distribution function does not satisfy the moment closure criterion and (iii) identification of the approximations to the material derivatives in the second-order distribution function that will correctly account for the difference in time scales between the first- and second-order fluxes. The first three terms of the Chapman–Enskog expansion, that defines the second-order distribution function, and the Bhatnagar–Gross–Krook model of the collision integral, form the basis of the BGK-Burnett equations. The entropy-consistent behaviour of the equations depend on the moment closure coefficients and the approximations to the material derivatives in the second-order fluxes. The requirement of moment closure alone, however, results in non-unique closure coefficients and a family of BGK-Burnett equations, from which an entropy-consistent set must be identified. From this family, two sets of BGK-Burnett equations have been considered, and this paper presents the details of the formulation of these two sets of equations, the identification of entropy-consistent approximations to the material derivatives by a novel entropy consistent relaxation technique, and shock structure computations in a monatomic hard sphere gas for a range of Mach numbers.

Generalization of the Hecke theorem for the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear collision integral in the Boltzmann equation are studied. Expansions in spherical Hermitean polynomials are used. It was shown [1] that the nonlinear matrix elements of the collision operator are related to each other by simple expressions, which are valid for arbitrary cross sections of particle interaction. The structure of the collision operator and the properties of the matrix elements are studied for the case when the interaction potential is spherically symmetric. In this case, the linear Boltzmann operator satisfies the Hecke theorem. The generalized Hecke theorem, from which it follows that many nonlinear matrix elements vanish, is proved with recurrence relations derived. It is shown that the generalized Hecke theorem is a consequence of the ordinary Hecke theorem.

Numerical modeling of the formation of steady-state nonequilibrium distributions of particles interacting through a power-law potential

The formation of a steady-state nonequilibrium distribution function of particles interacting through the repulsive potential U ~ α/rβ(1≤β≤4), which operates at an infinite range, is studied numerically. The collisional particle dynamics in such a system is investigated using a spatially homogeneous nonlinear collision integral in the Landau-Fokker-Planck form, which is a model Boltzmann collision integral for arbitrary potentials of interaction accompanied by little momentum transfer between particles in collisions. Numerical modeling is based on completely conservative difference schemes. It is shown that the principal condition for the existence of steady-state nonequilibrium distributions is the presence of a particle or an energy flux oriented in the proper manner in momentum space. A steady-state local distribution exists inside the momentum interval between the energy source and sink and has the form of a gradually decreasing function. Since a radical change in the distribution function under nonequilibrium conditions leads to an anomalous enhancement of the conduction of a medium and its emission characteristics, the results obtained can be used, e.g., to predict the behavior of semiconductors with an intrinsic or extrinsic conductivity under the action of particle fluxes or electromagnetic radiation.

Diffusion of charged particles in strong large-scale random and regular magnetic fields

The nonlinear collision integral for the Green’s function averaged over a random magnetic field is transformed using an iteration procedure taking account of the strong random scattering of particles on the correlation length of the random magnetic field. Under this transformation the regular magnetic field is assumed to be uniform at distances of the order of the correlation length. The single-particle Green’s functions of the scattered particles in the presence of a regular magnetic field are investigated. The transport coefficients are calculated taking account of the broadening of the cyclotron and Cherenkov resonances as a result of strong random scattering. The mean-free path lengths parallel and perpendicular to the regular magnetic field are found for a power-law spectrum of the random field. The analytical results obtained are compared with the experimental data on the transport ranges of solar and galactic cosmic rays in the interplanetary magnetic field. As a result, the conditions for the propagation of cosmic rays in the interplanetary space and a more accurate idea of the structure of the interplanetary magnetic field are determined.

Polynomial expansions for the isotropic Boltzmann equation and invariance of the collision integral with respect to the choice of basis functions

A new approach to the development of the nonlinear moment method for solving the Boltzmann equation is advanced. The approach is based on the invariance of the collision integral with respect to the choice of basis functions. It is shown that the matrix elements corresponding to the moments of the nonlinear collision integral are not independent, being related to each other by simple recursion relations. The latters are used for calculation of the matrix elements with large indices in the case of arbitrary power potentials. It is demonstrated that a numerical calculation procedure which enables constructing the distribution function up to eight–ten thermal velocities.

Nonlinear moment method for the isotropic Boltzmann equation and invariance of the collision integral

A new approach is proposed for the development of a nonlinear moment method of solving the Boltzmann equation. This approach is based on the principle of invariance of the collision integral with respect to the choice of basis functions. Sonine polynomials with a Maxwellian weighting function are taken as these basis functions for the velocity-isotropic Boltzmann equation. It is shown that for arbitrary interaction cross sections the matrix elements corresponding to the moments of the nonlinear collision integral are not independent but are coupled by simple recurrence formulas by means of which all the nonlinear matrix elements are expressed in terms of linear ones. As a result, a highly efficient numerical scheme is constructed for calculating the nonlinear matrix elements. The proposed approach opens up prospects for calculating relaxation processes at high velocities and also for solving more complex kinetic problems.

Theory of nonlinear interaction of particles and waves in an inverse plasma maser. Part 1. Collision integral

An expression is obtained for the collision integral describing the simultaneous interaction of plasma particles with resonant and non-resonant waves. It is shown that this collision integral is determined by two processes: a ‘direct’ nonlinear interaction of particles and waves, and the influence of the non-stationarity of the system. The expression for the nonlinear collision integral is found to be quite different from the expression for a quasi-linear collision integral; in particular, the nonlinear integral contains higher-order derivatives of the distribution function with respect to momentum than the quasi-linear one.