Bernie D. Shizgal

A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems

Abstract A new Gaussian quadrature procedure is developed for integrals of the form ∫ 0 ∞ e − y 2 y p ( y ) dy for p = 0, 1 and 2. Recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae. A comparison with the Gauss-Laguerre quadrature procedure is presented. Solutions of the chemical kinetic Boltzmann equation are obtained with a discrete ordinate method based on this Gaussian quadrature procedure. The results are compared with previous solutions obtained with a polynomial expansion method.

Nonthermal escape of the atmospheres of Venus, Earth, and Mars

Atmospheric loss from planetary atmo- spheres is an important geophysical problem with impli- cations for planetary evolution. This is a multidisci- plinary research field that requires an expertise in a wide range of subjects including statistical mechanics, fluid mechanics, plasma physics, collision theory, and surface science. This paper is a review of the current state of our understanding of atmospheric loss from the terrestrial planets. A detailed discussion is provided of the basic concepts required to understand the processes occurring in the high-altitude portion of a planetary atmosphere referred to as the exosphere. Light atomic species with sufficient translational energy can escape from an atmo- sphere. The translational energy required for escape could be thermal energy and proportional to the ambi- ent temperature or the result of some collisional pro- cesses energizing the species above thermal energies. These collisional processes, which include charge ex- change and dissociative recombination between ener- getic ions, neutrals, and electrons, are referred to as nonthermal escape processes. We highlight the similar- ities and differences in the important escape mechanisms on the terrestrial planets and comment on application of these mechanisms to evolutionary theories of the terres- trial atmospheres. The emphasis in this paper is directed toward the need to consider the exosphere as collisional.

Towards the resolution of the Gibbs phenomena

It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present paper introduces a new method for the resolution of the Gibbs phenomenon which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39 (1997) 644) based on Gegenbauer polynomials orthogonal with respect to weight function (1 - x2)λ-1/2. We refer to their approach as the direct method and to the new methodology as the inverse method. Both methods use the finite set of Fourier coefficients of some given function as input data in the re-expansion of the function in Gegenbauer polynomials or in other orthogonal basis sets. The finite partial sum of the new expansion provides a spectrally accurate approximation to the function. In the direct method, this requires that certain conditions are met concerning the parameter λ in the weight function, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of λ and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method.

A discrete ordinate method of solution of linear boundary value and eigenvalue problems

Abstract A discrete ordinate method is developed for the solution of linear differential equations. The method is based on a Gaussian quadrature procedure and is an extension of a discrete ordinate method used for the solution of integral equations. The present method is based on a representation of the derivative operator in a discrete ordinate basis. The method is applied to a number of problems with known solutions and is found to work extremely well.

Time dependent nucleation

Continuum approximations to the discrete birth and death equations for classical nucleation are investigated. The discrete equations are parametrized by rate coefficients αi and βi for a cluster of size i to lose or gain a monomer, respectively. The continuum equations considered for the distribution function f(x,t) of clusters containing x monomers at time t are all of the form of a Fokker–Planck equation: ∂f/∂t=∂/∂x[Bf eq∂(f/f eq)/∂x], where f eq(x) is the equilibrium distribution and B(x) is a diffusion coefficient. The dependence of B(x) on various continuum approximations to the rate coefficients is discussed at length. Three different forms of B(x) are considered; that used by Frenkel [Kinetic Theory of Liquids (Oxford, Oxford, 1946)], that suggested by Goodrich [Proc. R. Soc. London Ser. A 371, 167 (1964)], and a third form proposed here. Steady state distributions and time lags obtained from the continuous and discrete equations are compared. The time‐dependent Fokker–Planck equation is solved by ...

The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions

A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker–Planck equations. The Fokker–Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics. For this class of potentials, the ground state wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker–Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schro...

Vibrational nonequilibrium in a supersonic expansion with reaction: Application to O2–O

The hypersonic expansion of O2 through a nozzle is considered. The steady nonequilibrium vibrational distribution function of O2, and the nonequilibrium forward reaction rate coefficient for the dissociation of O2 are calculated theoretically. In the first instance, the vibrational relaxation of O2 in the absence of reaction is examined in the temperature range 500–2500 K. The master equation for the vibrational populations, coupled to the steady one‐dimensional conservation equations is solved numerically. The vibrational population obtained in this way, is compared to the distribution calculated using either a Treanor model or a Boltzmann distribution characterized by a vibrational temperature. The transition probabilities between O2 vibrational levels employed take into account the vibrational anharmonicity, and the anisotropic intermolecular potential. For the temperature range 2500–5000 K, the vibrational relaxation including dissociation and recombination is studied. The reactive probabilities that ...

Matrix elements of the Boltzmann collision operator for gas mixtures

Abstract The matrix elements of the collision operators that arise in the linearization of the Boltzman equations for gas mixtures are calculated. The matrix representatives of the operators with the Burnett functions (products of Laguerre polynomials and spherical harmonics) are evaluated for power law potentials. The final expressions for the matrix elements involve a small number of summations and for the hard sphere cross section, analytic expressions in terms of the mass ratios are obtained.

A collisional kinetic theory of a plane parallel evaporating planetary atmosphere

Abstract The departure from equilibrium in the exosphere of a planetary atmosphere owing to the loss of energetic atoms is studied with solutions of the Boltzmann equation. A rigorous collisional solution is obtained for a plane parallel model with a newly developed discrete ordinale method, and the decrease in the escape flux relative to Jeans flux is calculated together with density and temperature profiles. The escape of hydrogen and helium atoms from Earth and the escape of hydrogen atoms from Mars is considered. The results are compared with previous studies of the same effect. The departures from equilibrium are largest for the escape of a light species from an atmosphere with a heavy background gas such as for the escape of hydrogen atoms from Mars.

Eigenvalues of the boltzmann collision operator for binary gases: Relaxation of anisotropic distributions

Abstract The time eigenvalues of the Boltzmann collision operator for the hard-sphere cross section are calculated with a discrete ordinate method. The anisotropic part of the collision operator is considered in the present work. The approach to equilibrium of a gas initially non-maxwellian and anisotropic is considered.