Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

Willert, Jeffrey; Park, H.; Taitano, William

doi:10.13182/NSE14-131

Title: Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

Abstract

High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Lastly, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

Authors:

Willert, Jeffrey ^[1]; Park, H. ^[1]; Taitano, William ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Publication Date:: Sun Nov 01 00:00:00 EDT 2015

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1246339

Report Number(s):: LA-UR-14-28681
Journal ID: ISSN 0029-5639

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Nuclear Science and Engineering

Additional Journal Information:: Journal Volume: 181; Journal Issue: 3; Journal ID: ISSN 0029-5639

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Mathematics

Citation Formats


                    Willert, Jeffrey, Park, H., and Taitano, William. Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering.  United States: N. p., 2015. 
Web.  doi:10.13182/NSE14-131.

Copy to clipboard


                    Willert, Jeffrey, Park, H., & Taitano, William. Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering.  United States.  https://doi.org/10.13182/NSE14-131

Copy to clipboard


                    Willert, Jeffrey, Park, H., and Taitano, William. Sun .  
"Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering".  United States.  https://doi.org/10.13182/NSE14-131.  https://www.osti.gov/servlets/purl/1246339.

Copy to clipboard


                    
@article{osti_1246339,

  title        = {Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering},

  author       = {Willert, Jeffrey and Park, H. and Taitano, William},

  abstractNote = {High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Lastly, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.},

  doi          = {10.13182/NSE14-131},

  journal      = {Nuclear Science and Engineering},

  number       = 3,

  volume       = 181,

  place        = {United States},

  year         = {Sun Nov 01 00:00:00 EDT 2015},

  month        = {Sun Nov 01 00:00:00 EDT 2015}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.13182/NSE14-131

Other availability

Search WorldCat to find libraries that may hold this journal

Citation Metrics:

Cited by: 3 works

Citation information provided by
Web of Science

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

A comparison of acceleration methods for solving the neutron transport k -eigenvalue problem
journal, October 2014

Willert, Jeffrey; Park, H.; Knoll, D. A.
Journal of Computational Physics, Vol. 274
DOI: 10.1016/j.jcp.2014.06.044

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem
journal, April 2013

Calef, Matthew T.; Fichtl, Erin D.; Warsa, James S.
Journal of Computational Physics, Vol. 238
DOI: 10.1016/j.jcp.2012.12.024

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry
journal, February 2011

Knoll, D. A.; Park, H.; Smith, Kord
Nuclear Science and Engineering, Vol. 167, Issue 2
DOI: 10.13182/NSE09-75

Hybrid Krylov Methods for Nonlinear Systems of Equations
journal, May 1990

Brown, Peter N.; Saad, Youcef
SIAM Journal on Scientific and Statistical Computing, Vol. 11, Issue 3
DOI: 10.1137/0911026

A Functional Monte Carlo Method for k -Eigenvalue Problems
journal, June 2008

Larsen, Edward W.; Yang, Jinan
Nuclear Science and Engineering, Vol. 159, Issue 2
DOI: 10.13182/NSE07-92

Analytical benchmark test set for criticality code verification
journal, January 2003

Sood, Avneet; Forster, R. Arthur; Kent Parsons, D.
Progress in Nuclear Energy, Vol. 42, Issue 1
DOI: 10.1016/S0149-1970(02)00098-7

Works referencing / citing this record:

Using Anderson Acceleration to Accelerate the Convergence of Neutron Transport Calculations with Anisotropic Scattering
journal, November 2015

Willert, Jeffrey; Park, H.; Taitano, William
Nuclear Science and Engineering, Vol. 181, Issue 3
DOI: 10.13182/nse15-16

Similar Records in DOE PAGES and OSTI.GOV collections:

Applying Nonlinear Diffusion Acceleration to Fixed-Source Problems with Anisotropic Scattering - Paper 15

Conference Willert, Jeffrey ; Park, H. ; Taitano, William

Work over the last several years has demonstrated the impressive performance gains that can be attained by using Moment-Based Acceleration (MBA), or High-Order/Low-Order (HOLO), methods in a variety of transport applications. In the field of neutronics, Nonlinear Diffusion Acceleration (NDA) has been used to accelerate the solution to the fixed-source problem and the k-eigenvalue problem. These methods efficiently move a major portion of the computational work out of the full phase-space (high-order (HO)) domain into a reduced phase-space (low-order (LO)) setting. In previous implementations, NDA has only been applied to isotropic scattering problems. In this paper, we will extend thismore »« less
On the Degradation of the Effectiveness of Nonlinear Diffusion Acceleration with Parallel Block Jacobi Splitting

Journal Article Schunert, Sebastian ; Wang, Yaqi ; Ortensi, Javier ; ... - Transactions of the American Nuclear Society

MOOSE's radiation transport module Rattlesnake comprises a first order S N solver that can be accelerated with a flexible Nonlinear Diffusion Acceleration. At each NDA iteration, the streaming and collision operator of the S{sub N} equations are (approximately) inverted using a transport sweep. When executed on multiple processors parallel block Jacobi (PBJ) splitting is utilized lagging the processor sub-domain's incoming angular fluxes making the implemented method asynchronous, i.e. the convergence properties of NDA become dependent on the number of concurrent tasks and details of the partitioning. Within this work, we study the NDA convergence properties for the two-dimensional C5G7 benchmarkmore »« less
A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

Journal Article Willert, Jeffrey ; Park, H. ; Knoll, D. A. - Journal of Computational Physics

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton–Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (ormore »« less
https://doi.org/10.1016/J.JCP.2014.06.044
A hybrid approach to the neutron transport K-eigenvalue problem using NDA-based algorithms

Conference Willert, J. A. ; Kelley, C. T. ; Knoll, D. A. ; ...

In order to provide more physically accurate solutions to the neutron transport equation it has become increasingly popular to use Monte Carlo simulation to model nuclear reactor dynamics. These Monte Carlo methods can be extremely expensive, so we turn to a class of methods known as hybrid methods, which combine known deterministic and stochastic techniques to solve the transport equation. In our work, we show that we can simulate the action of a transport sweep using a Monte Carlo simulation in order to solve the k-eigenvalue problem. We'll accelerate the solution using nonlinear diffusion acceleration (NDA) as in [1,2]. Ourmore »« less
Efficient solutions to the NDA-NCA low-order eigenvalue problem

Conference Willert, J. A. ; Kelley, C. T.

Recent algorithmic advances combine moment-based acceleration and Jacobian-Free Newton-Krylov (JFNK) methods to accelerate the computation of the dominant eigenvalue in a k-eigenvalue calculation. In particular, NDA-NCA [1], builds a sequence of low-order (LO) diffusion-based eigenvalue problems in which the solution converges to the true eigenvalue solution. Within NDA-NCA, the solution to the LO k-eigenvalue problem is computed by solving a system of nonlinear equation using some variant of Newton's method. We show that we can speed up the solution to the LO problem dramatically by abandoning the JFNK method and exploiting the structure of the Jacobian matrix. (authors)

Similar Records

Title: Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

Abstract

Citation Formats

A comparison of acceleration methods for solving the neutron transport k -eigenvalue problem journal, October 2014

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem journal, April 2013

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry journal, February 2011

Hybrid Krylov Methods for Nonlinear Systems of Equations journal, May 1990

A Functional Monte Carlo Method for k -Eigenvalue Problems journal, June 2008

Analytical benchmark test set for criticality code verification journal, January 2003

Using Anderson Acceleration to Accelerate the Convergence of Neutron Transport Calculations with Anisotropic Scattering journal, November 2015

A comparison of acceleration methods for solving the neutron transport k -eigenvalue problem
journal, October 2014

Nonlinear Krylov acceleration applied to a discrete ordinates formulation of the k-eigenvalue problem
journal, April 2013

Application of the Jacobian-Free Newton-Krylov Method to Nonlinear Acceleration of Transport Source Iteration in Slab Geometry
journal, February 2011

Hybrid Krylov Methods for Nonlinear Systems of Equations
journal, May 1990

A Functional Monte Carlo Method for k -Eigenvalue Problems
journal, June 2008

Analytical benchmark test set for criticality code verification
journal, January 2003

Using Anderson Acceleration to Accelerate the Convergence of Neutron Transport Calculations with Anisotropic Scattering
journal, November 2015