# A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra

## Abstract

In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of amore »

- Authors:

- Texas A & M Univ., College Station, TX (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 775871

- Report Number(s):
- LA-13664-T

TRN: US0102017

- DOE Contract Number:
- W-7405-ENG-36

- Resource Type:
- Thesis/Dissertation

- Resource Relation:
- Other Information: TH: Thesis; Thesis information not supplied; PBD: 1 Nov 2000

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 22 GENERAL STUDIES OF NUCLEAR REACTORS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; ALGORITHMS; BOUNDARY LAYERS; CROSS SECTIONS; DIFFUSION; EIGENVALUES; NEUTRON TRANSPORT; PERFORMANCE; SCATTERING; TEST REACTORS; DISCRETE ORDINATE METHOD; MIXED OXIDE FUELS

### Citation Formats

```
Thompson, Kelly Glen.
```*A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra*. United States: N. p., 2000.
Web. doi:10.2172/775871.

```
Thompson, Kelly Glen.
```*A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra*. United States. doi:10.2172/775871.

```
Thompson, Kelly Glen. Wed .
"A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra". United States. doi:10.2172/775871. https://www.osti.gov/servlets/purl/775871.
```

```
@article{osti_775871,
```

title = {A Spatial Discretization Scheme for Solving the Transport Equation on Unstructured Grids of Polyhedra},

author = {Thompson, Kelly Glen},

abstractNote = {In this work, we develop a new spatial discretization scheme that may be used to numerically solve the neutron transport equation. This new discretization extends the family of corner balance spatial discretizations to include spatial grids of arbitrary polyhedra. This scheme enforces balance on subcell volumes called corners. It produces a lower triangular matrix for sweeping, is algebraically linear, is non-negative in a source-free absorber, and produces a robust and accurate solution in thick diffusive regions. Using an asymptotic analysis, we design the scheme so that in thick diffusive regions it will attain the same solution as an accurate polyhedral diffusion discretization. We then refine the approximations in the scheme to reduce numerical diffusion in vacuums, and we attempt to capture a second order truncation error. After we develop this Upstream Corner Balance Linear (UCBL) discretization we analyze its characteristics in several limits. We complete a full diffusion limit analysis showing that we capture the desired diffusion discretization in optically thick and highly scattering media. We review the upstream and linear properties of our discretization and then demonstrate that our scheme captures strictly non-negative solutions in source-free purely absorbing media. We then demonstrate the minimization of numerical diffusion of a beam and then demonstrate that the scheme is, in general, first order accurate. We also note that for slab-like problems our method actually behaves like a second-order method over a range of cell thicknesses that are of practical interest. We also discuss why our scheme is first order accurate for truly 3D problems and suggest changes in the algorithm that should make it a second-order accurate scheme. Finally, we demonstrate 3D UCBL's performance on several very different test problems. We show good performance in diffusive and streaming problems. We analyze truncation error in a 3D problem and demonstrate robustness in a coarsely discretized problem that contains sharp boundary layers. We also examine eigenvalue and fixed source problems with mixed-shape meshes, anisotropic scattering and multi-group cross sections. Finally, we simulate the MOX fuel assembly in the Advance Test Reactor.},

doi = {10.2172/775871},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Nov 01 00:00:00 EST 2000},

month = {Wed Nov 01 00:00:00 EST 2000}

}