# Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation

## Abstract

A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation. -- Highlights: •Wavelet angular discretisation used to solve transport equation. •Adaptive method developed for the wavelet discretisation. •Anisotropic angular resolution demonstrated through the adaptive method. •Adaptive method provides improvementsmore »

- Authors:

- Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, London, SW7 2AZ (United Kingdom)
- ANSWERS Software Service, AMEC, Kimmeridge House, Dorset Green Technology Park, Winfrith Newburgh, Dorchester, Dorset, DT2 8ZB (United Kingdom)
- AWE, Aldermaston, Reading, RG7 4PR (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22382176

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 281; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANISOTROPY; DIFFERENTIAL EQUATIONS; ERRORS; FINITE ELEMENT METHOD; NEUTRAL-PARTICLE TRANSPORT; RESOLUTION; SPHERICAL CONFIGURATION; TRANSPORT THEORY

### Citation Formats

```
Goffin, Mark A., E-mail: mark.a.goffin@gmail.com, Buchan, Andrew G., Dargaville, Steven, Pain, Christopher C., Smith, Paul N., and Smedley-Stevenson, Richard P.
```*Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation*. United States: N. p., 2015.
Web. doi:10.1016/J.JCP.2014.10.063.

```
Goffin, Mark A., E-mail: mark.a.goffin@gmail.com, Buchan, Andrew G., Dargaville, Steven, Pain, Christopher C., Smith, Paul N., & Smedley-Stevenson, Richard P.
```*Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation*. United States. doi:10.1016/J.JCP.2014.10.063.

```
Goffin, Mark A., E-mail: mark.a.goffin@gmail.com, Buchan, Andrew G., Dargaville, Steven, Pain, Christopher C., Smith, Paul N., and Smedley-Stevenson, Richard P. Thu .
"Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation". United States.
doi:10.1016/J.JCP.2014.10.063.
```

```
@article{osti_22382176,
```

title = {Goal-based angular adaptivity applied to a wavelet-based discretisation of the neutral particle transport equation},

author = {Goffin, Mark A., E-mail: mark.a.goffin@gmail.com and Buchan, Andrew G. and Dargaville, Steven and Pain, Christopher C. and Smith, Paul N. and Smedley-Stevenson, Richard P.},

abstractNote = {A method for applying goal-based adaptive methods to the angular resolution of the neutral particle transport equation is presented. The methods are applied to an octahedral wavelet discretisation of the spherical angular domain which allows for anisotropic resolution. The angular resolution is adapted across both the spatial and energy dimensions. The spatial domain is discretised using an inner-element sub-grid scale finite element method. The goal-based adaptive methods optimise the angular discretisation to minimise the error in a specific functional of the solution. The goal-based error estimators require the solution of an adjoint system to determine the importance to the specified functional. The error estimators and the novel methods to calculate them are described. Several examples are presented to demonstrate the effectiveness of the methods. It is shown that the methods can significantly reduce the number of unknowns and computational time required to obtain a given error. The novelty of the work is the use of goal-based adaptive methods to obtain anisotropic resolution in the angular domain for solving the transport equation. -- Highlights: •Wavelet angular discretisation used to solve transport equation. •Adaptive method developed for the wavelet discretisation. •Anisotropic angular resolution demonstrated through the adaptive method. •Adaptive method provides improvements in computational efficiency.},

doi = {10.1016/J.JCP.2014.10.063},

journal = {Journal of Computational Physics},

number = ,

volume = 281,

place = {United States},

year = {Thu Jan 15 00:00:00 EST 2015},

month = {Thu Jan 15 00:00:00 EST 2015}

}