# An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes

## Abstract

In this paper, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth–Fokker–Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, v _{th} (t), and explicitly considers the resulting inertial terms in the Fokker–Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker–Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic v _{th} -ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. Specifically, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1457268

- Alternate Identifier(s):
- OSTI ID: 1347623

- Report Number(s):
- LA-UR-15-27477

Journal ID: ISSN 0021-9991

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 318; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Conservative discretization; thermal velocity based adaptive grid; Fokker-Planck; Rosenbluth potentials; asymptotics

### Citation Formats

```
Taitano, William, Chacon, Luis, and Simakov, Andrei Nikolaevich.
```*An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes*. United States: N. p., 2016.
Web. doi:10.1016/j.jcp.2016.03.071.

```
Taitano, William, Chacon, Luis, & Simakov, Andrei Nikolaevich.
```*An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes*. United States. doi:10.1016/j.jcp.2016.03.071.

```
Taitano, William, Chacon, Luis, and Simakov, Andrei Nikolaevich. Mon .
"An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes". United States.
doi:10.1016/j.jcp.2016.03.071. https://www.osti.gov/servlets/purl/1457268.
```

```
@article{osti_1457268,
```

title = {An adaptive, conservative 0D-2V multispecies Rosenbluth–Fokker–Planck solver for arbitrarily disparate mass and temperature regimes},

author = {Taitano, William and Chacon, Luis and Simakov, Andrei Nikolaevich},

abstractNote = {In this paper, we propose an adaptive velocity-space discretization scheme for the multi-species, multidimensional Rosenbluth–Fokker–Planck (RFP) equation, which is exactly mass-, momentum-, and energy-conserving. Unlike most earlier studies, our approach normalizes the velocity-space coordinate to the temporally varying individual plasma species' local thermal velocity, vth (t), and explicitly considers the resulting inertial terms in the Fokker–Planck equation. Our conservation strategy employs nonlinear constraints to enforce discretely the conservation properties of these inertial terms and the Fokker–Planck collision operator. To deal with situations of extreme thermal velocity disparities among different species, we employ an asymptotic vth -ratio-based expansion of the Rosenbluth potentials that only requires the computation of several velocity-space integrals. Numerical examples demonstrate the favorable efficiency and accuracy properties of the scheme. Specifically, we show that the combined use of the velocity-grid adaptivity and asymptotic expansions delivers many orders-of-magnitude savings in mesh resolution requirements compared to a single, static uniform mesh.},

doi = {10.1016/j.jcp.2016.03.071},

journal = {Journal of Computational Physics},

number = C,

volume = 318,

place = {United States},

year = {Mon Apr 25 00:00:00 EDT 2016},

month = {Mon Apr 25 00:00:00 EDT 2016}

}

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