# Multigrid algorithms for solving the linear Boltzmann equation using first-order system least-squares finite element methods

## Abstract

Solving the linear Boltzmann equation in neutron scattering phenomena presents many challenges to standard numerical schemes in computational physics. For an SN discretization, the so-called ray effects pollute the numerical solution. This pollution can be viewed mathematically as ''contamination'' from a poorly chosen approximating basis set for the angle component of the discretization-i.e., collocation in angle is equivalent to discretization with delta basis functions, which form a poor approximating basis set. Fortunately, a PN discretization, which uses a better approximating basis set (i.e., spherical harmonics), eliminates these ray effects. Unfortunately, solving for the moments or PN equations is difficult. Moments couple strongly with each other, creating a strongly coupled system of partial differential equations (pde's) ; numerical algorithms for solving such strongly coupled systems are difficult to develop. In this paper, novel algorithms for solving this coupled system are presented. In particular, algorithms for solving the PN discretization of the linear Boltzmann equation using a first-order system least-squares (FOSLS) methodology (c.f. [1]) are presented.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- US Department of Energy (US)

- OSTI Identifier:
- 15006292

- Report Number(s):
- UCRL-JC-141742

TRN: US200407%%189

- DOE Contract Number:
- W-7405-ENG-48

- Resource Type:
- Conference

- Resource Relation:
- Conference: Nuclear Explosives Code Developers Conference, Oakland, CA (US), 10/23/2000--10/27/2000; Other Information: PBD: 13 Dec 2000

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; BOLTZMANN EQUATION; CONTAMINATION; FINITE ELEMENT METHOD; HARMONICS; NEUTRONS; NUCLEAR EXPLOSIVES; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; PHYSICS; POLLUTION; SCATTERING

### Citation Formats

```
Chang, B, and Lee, B.
```*Multigrid algorithms for solving the linear Boltzmann equation using first-order system least-squares finite element methods*. United States: N. p., 2000.
Web.

```
Chang, B, & Lee, B.
```*Multigrid algorithms for solving the linear Boltzmann equation using first-order system least-squares finite element methods*. United States.

```
Chang, B, and Lee, B. 2000.
"Multigrid algorithms for solving the linear Boltzmann equation using first-order system least-squares finite element methods". United States. https://www.osti.gov/servlets/purl/15006292.
```

```
@article{osti_15006292,
```

title = {Multigrid algorithms for solving the linear Boltzmann equation using first-order system least-squares finite element methods},

author = {Chang, B and Lee, B},

abstractNote = {Solving the linear Boltzmann equation in neutron scattering phenomena presents many challenges to standard numerical schemes in computational physics. For an SN discretization, the so-called ray effects pollute the numerical solution. This pollution can be viewed mathematically as ''contamination'' from a poorly chosen approximating basis set for the angle component of the discretization-i.e., collocation in angle is equivalent to discretization with delta basis functions, which form a poor approximating basis set. Fortunately, a PN discretization, which uses a better approximating basis set (i.e., spherical harmonics), eliminates these ray effects. Unfortunately, solving for the moments or PN equations is difficult. Moments couple strongly with each other, creating a strongly coupled system of partial differential equations (pde's) ; numerical algorithms for solving such strongly coupled systems are difficult to develop. In this paper, novel algorithms for solving this coupled system are presented. In particular, algorithms for solving the PN discretization of the linear Boltzmann equation using a first-order system least-squares (FOSLS) methodology (c.f. [1]) are presented.},

doi = {},

url = {https://www.osti.gov/biblio/15006292},
journal = {},

number = ,

volume = ,

place = {United States},

year = {2000},

month = {12}

}