# A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas

## Abstract

A high-order finite volume algorithm is developed for the Fokker-Planck Operator (FPO) describing Coulomb collisions in strongly magnetized plasmas. The algorithm is based on a general fourth-order reconstruction scheme for an unstructured grid in the velocity space spanned by parallel velocity and magnetic moment. The method provides density conservation and high-order-accurate evaluation of the FPO independent of the choice of the velocity coordinates. As an example, a linearized FPO in constant-of-motion coordinates, i.e. the total energy and the magnetic moment, is developed using the present algorithm combined with a cut-cell merging procedure. Numerical tests include the Spitzer thermalization problem and the return to isotropy for distributions initialized with velocity space loss cones. Utilization of the method for a nonlinear FPO is straightforward but requires evaluation of the Rosenbluth potentials.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 940853

- Report Number(s):
- UCRL-JRNL-230192

Journal ID: ISSN 0021-9991; JCTPAH; TRN: US0807225

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics, vol. 227, N/A, April 15, 2008, pp. 7192; Journal Volume: 227

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION; 99 GENERAL AND MISCELLANEOUS; ALGORITHMS; EVALUATION; ISOTROPY; LOSS CONE; MAGNETIC MOMENTS; THERMALIZATION; VELOCITY

### Citation Formats

```
Xiong, Z, Cohen, R H, Rognlien, T D, and Xu, X Q.
```*A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas*. United States: N. p., 2007.
Web.

```
Xiong, Z, Cohen, R H, Rognlien, T D, & Xu, X Q.
```*A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas*. United States.

```
Xiong, Z, Cohen, R H, Rognlien, T D, and Xu, X Q. Wed .
"A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas". United States.
doi:. https://www.osti.gov/servlets/purl/940853.
```

```
@article{osti_940853,
```

title = {A High-Order Finite-Volume Algorithm for Fokker-Planck Collisions in Magnetized Plasmas},

author = {Xiong, Z and Cohen, R H and Rognlien, T D and Xu, X Q},

abstractNote = {A high-order finite volume algorithm is developed for the Fokker-Planck Operator (FPO) describing Coulomb collisions in strongly magnetized plasmas. The algorithm is based on a general fourth-order reconstruction scheme for an unstructured grid in the velocity space spanned by parallel velocity and magnetic moment. The method provides density conservation and high-order-accurate evaluation of the FPO independent of the choice of the velocity coordinates. As an example, a linearized FPO in constant-of-motion coordinates, i.e. the total energy and the magnetic moment, is developed using the present algorithm combined with a cut-cell merging procedure. Numerical tests include the Spitzer thermalization problem and the return to isotropy for distributions initialized with velocity space loss cones. Utilization of the method for a nonlinear FPO is straightforward but requires evaluation of the Rosenbluth potentials.},

doi = {},

journal = {Journal of Computational Physics, vol. 227, N/A, April 15, 2008, pp. 7192},

number = ,

volume = 227,

place = {United States},

year = {Wed Apr 18 00:00:00 EDT 2007},

month = {Wed Apr 18 00:00:00 EDT 2007}

}