# A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework

## Abstract

In this paper, we present a domain decomposition algorithm to accelerate the solution of Eulerian-type discretizations of the linear, steady-state Vlasov equation. The steady-state solver then forms a key component in the implementation of fully implicit or nearly fully implicit temporal integrators for the nonlinear Vlasov--Poisson system. The solver relies on a particular decomposition of phase space that enables the use of sweeping techniques commonly used in radiation transport applications. The original linear system for the phase space unknowns is then replaced by a smaller linear system involving only unknowns on the boundary between subdomains, which can then be solved efficiently with Krylov methods such as GMRES. Steady-state solves are combined to form an implicit Runge--Kutta time integrator, and the Vlasov equation is coupled self-consistently to the Poisson equation via a linearized procedure or a nonlinear fixed-point method for the electric field. Finally, numerical results for standard test problems demonstrate the efficiency of the domain decomposition approach when compared to the direct application of an iterative solver to the original linear system.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); ORNL Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1435515

- Report Number(s):
- LA-UR-17-24638

Journal ID: ISSN 1064-8275

- Grant/Contract Number:
- AC52-06NA25396; AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 2; Journal ID: ISSN 1064-8275

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Vlasov-Poisson; implicit time integration; domain decomposition

### Citation Formats

```
Garrett, C. Kristopher, and Hauck, Cory D..
```*A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework*. United States: N. p., 2018.
Web. doi:10.1137/17M1134184.

```
Garrett, C. Kristopher, & Hauck, Cory D..
```*A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework*. United States. doi:10.1137/17M1134184.

```
Garrett, C. Kristopher, and Hauck, Cory D.. Thu .
"A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework". United States.
doi:10.1137/17M1134184.
```

```
@article{osti_1435515,
```

title = {A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework},

author = {Garrett, C. Kristopher and Hauck, Cory D.},

abstractNote = {In this paper, we present a domain decomposition algorithm to accelerate the solution of Eulerian-type discretizations of the linear, steady-state Vlasov equation. The steady-state solver then forms a key component in the implementation of fully implicit or nearly fully implicit temporal integrators for the nonlinear Vlasov--Poisson system. The solver relies on a particular decomposition of phase space that enables the use of sweeping techniques commonly used in radiation transport applications. The original linear system for the phase space unknowns is then replaced by a smaller linear system involving only unknowns on the boundary between subdomains, which can then be solved efficiently with Krylov methods such as GMRES. Steady-state solves are combined to form an implicit Runge--Kutta time integrator, and the Vlasov equation is coupled self-consistently to the Poisson equation via a linearized procedure or a nonlinear fixed-point method for the electric field. Finally, numerical results for standard test problems demonstrate the efficiency of the domain decomposition approach when compared to the direct application of an iterative solver to the original linear system.},

doi = {10.1137/17M1134184},

journal = {SIAM Journal on Scientific Computing},

number = 2,

volume = 40,

place = {United States},

year = {Thu Apr 05 00:00:00 EDT 2018},

month = {Thu Apr 05 00:00:00 EDT 2018}

}