# A Numerical Instability in an ADI Algorithm for Gyrokinetics

## Abstract

We explore the implementation of an Alternating Direction Implicit (ADI) algorithm for a gyrokinetic plasma problem and its resulting numerical stability properties. This algorithm, which uses a standard ADI scheme to divide the field solve from the particle distribution function advance, has previously been found to work well for certain plasma kinetic problems involving one spatial and two velocity dimensions, including collisions and an electric field. However, for the gyrokinetic problem we find a severe stability restriction on the time step. Furthermore, we find that this numerical instability limitation also affects some other algorithms, such as a partially implicit Adams-Bashforth algorithm, where the parallel motion operator v{sub {parallel}} {partial_derivative}/{partial_derivative}z is treated implicitly and the field terms are treated with an Adams-Bashforth explicit scheme. Fully explicit algorithms applied to all terms can be better at long wavelengths than these ADI or partially implicit algorithms.

- Authors:

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab., Princeton, NJ (US)

- Sponsoring Org.:
- USDOE Office of Science (SC) (US)

- OSTI Identifier:
- 836494

- Report Number(s):
- PPPL-4034

TRN: US0500615

- DOE Contract Number:
- AC02-76CH03073

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 17 Dec 2004

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; ALGORITHMS; DIMENSIONS; DISTRIBUTION FUNCTIONS; ELECTRIC FIELDS; IMPLEMENTATION; INSTABILITY; KINETICS; PLASMA; STABILITY; VELOCITY; WAVELENGTHS; COMPUTATIONAL PHYSICS; GYROKINETIC EQUATIONS; NUMERICAL METHODS

### Citation Formats

```
E.A. Belli, and G.W. Hammett.
```*A Numerical Instability in an ADI Algorithm for Gyrokinetics*. United States: N. p., 2004.
Web. doi:10.2172/836494.

```
E.A. Belli, & G.W. Hammett.
```*A Numerical Instability in an ADI Algorithm for Gyrokinetics*. United States. doi:10.2172/836494.

```
E.A. Belli, and G.W. Hammett. Fri .
"A Numerical Instability in an ADI Algorithm for Gyrokinetics". United States. doi:10.2172/836494. https://www.osti.gov/servlets/purl/836494.
```

```
@article{osti_836494,
```

title = {A Numerical Instability in an ADI Algorithm for Gyrokinetics},

author = {E.A. Belli and G.W. Hammett},

abstractNote = {We explore the implementation of an Alternating Direction Implicit (ADI) algorithm for a gyrokinetic plasma problem and its resulting numerical stability properties. This algorithm, which uses a standard ADI scheme to divide the field solve from the particle distribution function advance, has previously been found to work well for certain plasma kinetic problems involving one spatial and two velocity dimensions, including collisions and an electric field. However, for the gyrokinetic problem we find a severe stability restriction on the time step. Furthermore, we find that this numerical instability limitation also affects some other algorithms, such as a partially implicit Adams-Bashforth algorithm, where the parallel motion operator v{sub {parallel}} {partial_derivative}/{partial_derivative}z is treated implicitly and the field terms are treated with an Adams-Bashforth explicit scheme. Fully explicit algorithms applied to all terms can be better at long wavelengths than these ADI or partially implicit algorithms.},

doi = {10.2172/836494},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2004},

month = {12}

}