# Three-dimensional poisson solver for a charged beam with large aspect ratio in a conducting pipe

## Abstract

In this paper, we present a three-dimensional Poisson equation solver for the electrostatic potential of a charged beam with large longitudinal to transverse aspect ratio in a straight and a bent conducting pipe with open-end boundary conditions. In this solver, we have used a Hermite-Gaussian series to represent the longitudinal spatial dependence of the charge density and the electric potential. Using the Hermite-Gaussian approximation, the original three-dimensional Poisson equation has been reduced into a group of coupled two-dimensional partial differential equations with the coupling strength proportional to the inverse square of the longitudinal-to-transverse aspect ratio. For a large aspect ratio, the coupling is weak. These two-dimensional partial differential equations can be solved independently using an iterative approach. The iterations converge quickly due to the large aspect ration of the beam. For a transverse round conducting pipe, the two-dimensional Poisson equation is solved using a Bessel function approximation and a Fourier function approximation. The three-dimensional Poisson solver can have important applications in the study of the space-charge effects in the high intensity proton storage ring accelerator or induction linear accelerator for heavy ion fusion where the ration of bunch length to the transverse size is large.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Director. Office of Science. Office of High Energy and Nuclear Physics. Division of High Energy Physics, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing Project (US)

- OSTI Identifier:
- 837240

- Report Number(s):
- LBNL-54282

R&D Project: 453401; TRN: US0500936

- DOE Contract Number:
- AC03-76SF00098

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Computer Physics Communications; Journal Volume: 160; Journal Issue: 2; Other Information: Submitted to Computer Physics Communications: Volume 160, No.2; Journal Publication Date: 07/01/2004; PBD: 8 Jan 2004

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 43 PARTICLE ACCELERATORS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ACCELERATORS; ASPECT RATIO; BESSEL FUNCTIONS; BOUNDARY CONDITIONS; CHARGE DENSITY; ELECTRIC POTENTIAL; ELECTROSTATICS; HEAVY IONS; INDUCTION; LINEAR ACCELERATORS; PARTIAL DIFFERENTIAL EQUATIONS; POISSON EQUATION; PROTONS; SPACE CHARGE; SPACE DEPENDENCE; STORAGE RINGS; POISSON SOLVER HERMITE-GAUSSIAN APPROXIMATION LARGE ASPECT RATIO OPEN BOUNDARY CONDITIONS

### Citation Formats

```
Qiang, Ji, and Gluckstern, Robert L.
```*Three-dimensional poisson solver for a charged beam with large aspect ratio in a conducting pipe*. United States: N. p., 2004.
Web. doi:10.1016/j.cpc.2004.03.002.

```
Qiang, Ji, & Gluckstern, Robert L.
```*Three-dimensional poisson solver for a charged beam with large aspect ratio in a conducting pipe*. United States. doi:10.1016/j.cpc.2004.03.002.

```
Qiang, Ji, and Gluckstern, Robert L. Thu .
"Three-dimensional poisson solver for a charged beam with large aspect ratio in a conducting pipe". United States. doi:10.1016/j.cpc.2004.03.002. https://www.osti.gov/servlets/purl/837240.
```

```
@article{osti_837240,
```

title = {Three-dimensional poisson solver for a charged beam with large aspect ratio in a conducting pipe},

author = {Qiang, Ji and Gluckstern, Robert L.},

abstractNote = {In this paper, we present a three-dimensional Poisson equation solver for the electrostatic potential of a charged beam with large longitudinal to transverse aspect ratio in a straight and a bent conducting pipe with open-end boundary conditions. In this solver, we have used a Hermite-Gaussian series to represent the longitudinal spatial dependence of the charge density and the electric potential. Using the Hermite-Gaussian approximation, the original three-dimensional Poisson equation has been reduced into a group of coupled two-dimensional partial differential equations with the coupling strength proportional to the inverse square of the longitudinal-to-transverse aspect ratio. For a large aspect ratio, the coupling is weak. These two-dimensional partial differential equations can be solved independently using an iterative approach. The iterations converge quickly due to the large aspect ration of the beam. For a transverse round conducting pipe, the two-dimensional Poisson equation is solved using a Bessel function approximation and a Fourier function approximation. The three-dimensional Poisson solver can have important applications in the study of the space-charge effects in the high intensity proton storage ring accelerator or induction linear accelerator for heavy ion fusion where the ration of bunch length to the transverse size is large.},

doi = {10.1016/j.cpc.2004.03.002},

journal = {Computer Physics Communications},

number = 2,

volume = 160,

place = {United States},

year = {Thu Jan 08 00:00:00 EST 2004},

month = {Thu Jan 08 00:00:00 EST 2004}

}