# ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation

## Abstract

Resolving numerically Vlasov–Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65–67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuationmore »

- Authors:

- Institut d'Astrophysique de Paris, CNRS UMR 7095 and UPMC, 98bis, bd Arago, F-75014 Paris (France)
- (Japan)

- Publication Date:

- OSTI Identifier:
- 22572354

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 321; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CHAOS THEORY; COSMOLOGY; EQUATIONS OF MOTION; HAMILTONIANS; LAGRANGIAN FUNCTION; NONLUMINOUS MATTER; PHASE SPACE; POISSON EQUATION; POTENTIALS; SHEETS; SIMULATION

### Citation Formats

```
Sousbie, Thierry, E-mail: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 113-0033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Colombi, Stéphane, E-mail: colombi@iap.fr, and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502.
```*ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation*. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.05.048.

```
Sousbie, Thierry, E-mail: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 113-0033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Colombi, Stéphane, E-mail: colombi@iap.fr, & Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502.
```*ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation*. United States. doi:10.1016/J.JCP.2016.05.048.

```
Sousbie, Thierry, E-mail: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 113-0033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Colombi, Stéphane, E-mail: colombi@iap.fr, and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502. Thu .
"ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation". United States.
doi:10.1016/J.JCP.2016.05.048.
```

```
@article{osti_22572354,
```

title = {ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation},

author = {Sousbie, Thierry, E-mail: tsousbie@gmail.com and Department of Physics, The University of Tokyo, Tokyo 113-0033 and Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033 and Colombi, Stéphane, E-mail: colombi@iap.fr and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502},

abstractNote = {Resolving numerically Vlasov–Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65–67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a “warm” dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code.},

doi = {10.1016/J.JCP.2016.05.048},

journal = {Journal of Computational Physics},

number = ,

volume = 321,

place = {United States},

year = {Thu Sep 15 00:00:00 EDT 2016},

month = {Thu Sep 15 00:00:00 EDT 2016}

}