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

## Abstract

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 asmore »

- Authors:

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

- Publication Date:

- OSTI Identifier:
- 22314879

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 270; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 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

### Citation Formats

```
Gamba, Irene M., ICES, The University of Texas at Austin, 201 E. 24th St., Stop C0200, Austin, TX 78712, and Haack, Jeffrey R.
```*A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.03.035.

```
Gamba, Irene M., ICES, The University of Texas at Austin, 201 E. 24th St., Stop C0200, Austin, TX 78712, & Haack, Jeffrey R.
```*A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit*. United States. doi:10.1016/J.JCP.2014.03.035.

```
Gamba, Irene M., ICES, The University of Texas at Austin, 201 E. 24th St., Stop C0200, Austin, TX 78712, and Haack, Jeffrey R. Fri .
"A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit". United States.
doi:10.1016/J.JCP.2014.03.035.
```

```
@article{osti_22314879,
```

title = {A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit},

author = {Gamba, Irene M. and ICES, The University of Texas at Austin, 201 E. 24th St., Stop C0200, Austin, TX 78712 and Haack, Jeffrey R.},

abstractNote = {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.},

doi = {10.1016/J.JCP.2014.03.035},

journal = {Journal of Computational Physics},

number = ,

volume = 270,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}