A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit

Journal Article · Fri Aug 01 04:00:00 EDT 2014 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2014.03.035· OSTI ID:22314879

Gamba, Irene M. ^[1]; Haack, Jeffrey R. ^[1]

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1. Department of Mathematics, The University of Texas at Austin, 2515 Speedway, Stop C1200, Austin, TX 78712 (United States)

We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation.

• OSTI ID: 22314879
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 270; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANISOTROPY
APPROXIMATIONS
ASYMPTOTIC SOLUTIONS
BOLTZMANN EQUATION
COLLISIONS
COMPARATIVE EVALUATIONS
CONVERGENCE
CROSS SECTIONS
ENTROPY
FOKKER-PLANCK EQUATION
FOURIER TRANSFORMATION
MATHEMATICAL OPERATORS
NUMERICAL ANALYSIS
RUTHERFORD SCATTERING
SINGULARITY