# Convergence of Spectral Discretizations of the Vlasov--Poisson System

## Abstract

Here we prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in detail for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence under suitable regularity assumptions on the exact solution.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Univ. degli Studi di Modena e Reggio-Emilia, Modena (Italy). Dipartimento di Scienze Fisiche, Informatiche e Matematiche

- Publication Date:

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

- Sponsoring Org.:
- USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1400123

- Report Number(s):
- LA-UR-16-22601

Journal ID: ISSN 0036-1429

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- SIAM Journal on Numerical Analysis

- Additional Journal Information:
- Journal Volume: 55; Journal Issue: 5; Journal ID: ISSN 0036-1429

- Publisher:
- Society for Industrial and Applied Mathematics

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Mathematics; Hermite spectral method, Legendre spectral method, Vlasov equation, Vlasov-Poisson system

### Citation Formats

```
Manzini, G., Funaro, D., and Delzanno, G. L.
```*Convergence of Spectral Discretizations of the Vlasov--Poisson System*. United States: N. p., 2017.
Web. doi:10.1137/16M1076848.

```
Manzini, G., Funaro, D., & Delzanno, G. L.
```*Convergence of Spectral Discretizations of the Vlasov--Poisson System*. United States. doi:10.1137/16M1076848.

```
Manzini, G., Funaro, D., and Delzanno, G. L. Tue .
"Convergence of Spectral Discretizations of the Vlasov--Poisson System". United States.
doi:10.1137/16M1076848.
```

```
@article{osti_1400123,
```

title = {Convergence of Spectral Discretizations of the Vlasov--Poisson System},

author = {Manzini, G. and Funaro, D. and Delzanno, G. L.},

abstractNote = {Here we prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in detail for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence under suitable regularity assumptions on the exact solution.},

doi = {10.1137/16M1076848},

journal = {SIAM Journal on Numerical Analysis},

number = 5,

volume = 55,

place = {United States},

year = {Tue Sep 26 00:00:00 EDT 2017},

month = {Tue Sep 26 00:00:00 EDT 2017}

}