# Flux limiting nature`s own way -- A new method for numerical solution of the transport equation

## Abstract

The transport equation may be solved by expanding it in spherical harmonics, Y{sub lm}, and truncating the resultant infinite set of equations at some finite order L. This procedure leaves the (L + 1)th order moments which appear in the Lth order equation undetermined, and the standard procedure for obtaining a closed set of equations has been to set all the (L + 1)th order moments to zero. It has been shown here that this procedure actually violates the apriori knowledge that one is solving for the moments of a probability measure on the unit sphere. Using the theory of moments of a probability measure on the unit sphere. Using the theory of moments as discussed above, the (L + 1)th order moments can be chosen in accordance with apriori knowledge. The resultant truncated set of equations has properties much truer to the original transport equation than the usual set obtained by setting the (L + 1)th order moments to zero. In particular the truncated set of equations gets the solution of the transport equation exactly right in both the diffusion limit and the free streaming limit. Furthermore, this has been achieved by merely truncating the set of equations properlymore »

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 104974

- Report Number(s):
- UCRL-78378

ON: DE95017708; TRN: 95:006936

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

- Resource Type:
- Technical Report

- Resource Relation:
- Other Information: PBD: 29 Jul 1976

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; 99 MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS; TRANSPORT THEORY; MOMENTS METHOD; SPHERICAL HARMONICS; NUMERICAL SOLUTION

### Citation Formats

```
Kershaw, D S.
```*Flux limiting nature`s own way -- A new method for numerical solution of the transport equation*. United States: N. p., 1976.
Web. doi:10.2172/104974.

```
Kershaw, D S.
```*Flux limiting nature`s own way -- A new method for numerical solution of the transport equation*. United States. doi:10.2172/104974.

```
Kershaw, D S. Thu .
"Flux limiting nature`s own way -- A new method for numerical solution of the transport equation". United States. doi:10.2172/104974. https://www.osti.gov/servlets/purl/104974.
```

```
@article{osti_104974,
```

title = {Flux limiting nature`s own way -- A new method for numerical solution of the transport equation},

author = {Kershaw, D S},

abstractNote = {The transport equation may be solved by expanding it in spherical harmonics, Y{sub lm}, and truncating the resultant infinite set of equations at some finite order L. This procedure leaves the (L + 1)th order moments which appear in the Lth order equation undetermined, and the standard procedure for obtaining a closed set of equations has been to set all the (L + 1)th order moments to zero. It has been shown here that this procedure actually violates the apriori knowledge that one is solving for the moments of a probability measure on the unit sphere. Using the theory of moments of a probability measure on the unit sphere. Using the theory of moments as discussed above, the (L + 1)th order moments can be chosen in accordance with apriori knowledge. The resultant truncated set of equations has properties much truer to the original transport equation than the usual set obtained by setting the (L + 1)th order moments to zero. In particular the truncated set of equations gets the solution of the transport equation exactly right in both the diffusion limit and the free streaming limit. Furthermore, this has been achieved by merely truncating the set of equations properly and not by any ad hoc changes in the basic equations as is the case in the approaches that use flux limiters.},

doi = {10.2172/104974},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1976},

month = {7}

}