# Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions

## Abstract

The common definition of the diffusion coeffcient as the inverse of three times the transport cross section is not compat- ible with voids. Morel introduced a non-local tensor diffusion coeffcient that remains finite in voids[1]. It can be obtained by solving an auxiliary transport problem without scattering or fission. Larsen and Trahan successfully applied this diffusion coeffcient for enhancing the accuracy of diffusion solutions of very high temperature reactor (VHTR) problems that feature large, optically thin channels in the z-direction [2]. It is demonstrated that a significant reduction of error can be achieved in particular in the optically thin region. Along the same line of thought, non-local diffusion tensors are applied modeling the TREAT reactor confirming the findings of Larsen and Trahan [3]. Previous work of the authors have introduced a flexible Nonlinear-Diffusion Acceleration (NDA) method for the first order S N equations discretized with the discontinuous finite element method (DFEM), [4], [5], [6]. This NDA method uses a scalar diffusion coeffcient in the low-order system that is obtained as the flux weighted average of the inverse transport cross section. Hence, it su?ers from very large and potentially unbounded diffusion coeffcients in the low order problem. However, it was notedmore »

- Authors:

- Publication Date:

- Research Org.:
- Idaho National Lab. (INL), Idaho Falls, ID (United States)

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE)

- OSTI Identifier:
- 1363907

- Report Number(s):
- INL/CON-16-38952

- DOE Contract Number:
- DE-AC07-05ID14517

- Resource Type:
- Conference

- Resource Relation:
- Conference: 2016 ANS Winter Meeting and Nuclear Technology Expo, Las Vegas, Nevada, November 6–10, 2016

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Nonlinear Diffusion Acceleration; Radiation Transport; Rattlesnake

### Citation Formats

```
Schunert, Sebastian, Hammer, Hans, Lou, Jijie, Wang, Yaqi, Ortensi, Javier, Gleicher, Frederick, Baker, Benjamin, DeHart, Mark, and Martineau, Richard.
```*Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions*. United States: N. p., 2016.
Web.

```
Schunert, Sebastian, Hammer, Hans, Lou, Jijie, Wang, Yaqi, Ortensi, Javier, Gleicher, Frederick, Baker, Benjamin, DeHart, Mark, & Martineau, Richard.
```*Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions*. United States.

```
Schunert, Sebastian, Hammer, Hans, Lou, Jijie, Wang, Yaqi, Ortensi, Javier, Gleicher, Frederick, Baker, Benjamin, DeHart, Mark, and Martineau, Richard. Tue .
"Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions". United States. https://www.osti.gov/servlets/purl/1363907.
```

```
@article{osti_1363907,
```

title = {Using Directional Diffusion Coefficients for Nonlinear Diffusion Acceleration of the First Order SN Equations in Near-Void Regions},

author = {Schunert, Sebastian and Hammer, Hans and Lou, Jijie and Wang, Yaqi and Ortensi, Javier and Gleicher, Frederick and Baker, Benjamin and DeHart, Mark and Martineau, Richard},

abstractNote = {The common definition of the diffusion coeffcient as the inverse of three times the transport cross section is not compat- ible with voids. Morel introduced a non-local tensor diffusion coeffcient that remains finite in voids[1]. It can be obtained by solving an auxiliary transport problem without scattering or fission. Larsen and Trahan successfully applied this diffusion coeffcient for enhancing the accuracy of diffusion solutions of very high temperature reactor (VHTR) problems that feature large, optically thin channels in the z-direction [2]. It is demonstrated that a significant reduction of error can be achieved in particular in the optically thin region. Along the same line of thought, non-local diffusion tensors are applied modeling the TREAT reactor confirming the findings of Larsen and Trahan [3]. Previous work of the authors have introduced a flexible Nonlinear-Diffusion Acceleration (NDA) method for the first order S N equations discretized with the discontinuous finite element method (DFEM), [4], [5], [6]. This NDA method uses a scalar diffusion coeffcient in the low-order system that is obtained as the flux weighted average of the inverse transport cross section. Hence, it su?ers from very large and potentially unbounded diffusion coeffcients in the low order problem. However, it was noted that the choice of the diffusion coeffcient does not influence consistency of the method at convergence and hence the di?usion coeffcient is essentially a free parameter. The choice of the di?usion coeffcient does, however, affect the convergence behavior of the nonlinear di?usion iterations. Within this work we use Morel’s non-local di?usion coef- ficient in the aforementioned NDA formulation in lieu of the flux weighted inverse of three times the transport cross section. The goal of this paper is to demonstrate that significant en- hancement of the spectral properties of NDA can be achieved in near void regions. For testing the spectral properties of the NDA with non-local diffusion coeffcients, the periodical horizontal interface problem is used [7]. This problem consists of alternating stripes of optically thin and thick materials both of which feature scattering ratios close to unity.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2016},

month = {11}

}