# Metriplectic integrators for the Landau collision operator

## Abstract

Here, we present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations, in general, independently of the mesh configuration.

- Authors:

- Max-Planck-Institut fur Plasmaphysik, Garching (Deutschland); Technische Univ. Munchen, Garching (Deutschland)
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Publication Date:

- Research Org.:
- Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1411216

- Alternate Identifier(s):
- OSTI ID: 1395914

- Grant/Contract Number:
- 708124; AC02-09CH11466

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 24; Journal Issue: 10; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

### Citation Formats

```
Kraus, Michael, and Hirvijoki, Eero.
```*Metriplectic integrators for the Landau collision operator*. United States: N. p., 2017.
Web. doi:10.1063/1.4998610.

```
Kraus, Michael, & Hirvijoki, Eero.
```*Metriplectic integrators for the Landau collision operator*. United States. doi:10.1063/1.4998610.

```
Kraus, Michael, and Hirvijoki, Eero. Mon .
"Metriplectic integrators for the Landau collision operator". United States.
doi:10.1063/1.4998610. https://www.osti.gov/servlets/purl/1411216.
```

```
@article{osti_1411216,
```

title = {Metriplectic integrators for the Landau collision operator},

author = {Kraus, Michael and Hirvijoki, Eero},

abstractNote = {Here, we present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations, in general, independently of the mesh configuration.},

doi = {10.1063/1.4998610},

journal = {Physics of Plasmas},

number = 10,

volume = 24,

place = {United States},

year = {Mon Oct 02 00:00:00 EDT 2017},

month = {Mon Oct 02 00:00:00 EDT 2017}

}

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