# Newton-Krylov methods for the solution of the k-eigenvalue problem in multigroup neutronics calculations.

### Overview

In this work we propose using Newton's method, specifically the inexact Newton-GMRES formulation, to solve the k-eigenvalue problem in both transport and diffusion neutronics problems. This is achieved by choosing a nonlinear function whose roots are the eigenpairs of the k-eigenvalue calculation and then using Newton's method to solve the nonlinear system. The exibility resulting from the use of a Krylov subspace method to solve the linear Newton step can be further extended via the use of the Jacobian-Free Newton-Krylov (JFNK) approximation, which requires no knowledge of the system's Jacobian; instead only the ability to evaluate the system residual is necessary.;For the diffusion approximation, the nonlinear function is written in the form of the generalized eigenvalue problem and a set of preconditioners is developed and applied to the GMRES iterations that are used to solve the linearized Newton problem. Most of the developed methods can be implemented as either Newton-Krylov (NK) methods, where the Jacobian-vector product is formed using the explicitly constructed Jacobian, or via the JFNK approximation, where a finite-difference perturbation is used to approximate the Jacobian-vector product. One particularly effective preconditioning option comprises the use of the standard power iteration to precondition the GMRES iteration on either the right or the left. Pre-conditioning on the left, denoted JFNK(PI), results in a modified nonlinear system whose implementation only requires the ability to perform a single traditional outer iteration, making this approach relatively simple to wrap around an existing diffusion theory k-eigenvalue problem solver.;Similar methods were developed for transport theory, cast using an operator notation that greatly simplifies their presentation. All of the nonlinear functions developed are written in terms of a generic fixed-point iteration, with a number of specific fixed-point formulations considered. Each fixed-point scheme represents a viable k-eigenvalue problem solution method, with two of the techniques corresponding to traditionally used iterative schemes. The new methods developed can also be wrapped around existing software in most instances, simplifying the implementation process. Ultimately it is seen that the most effective of the Newton formulations in transport theory is wrapped around a k-eigenvalue formulation that is a very special instance of traditional methods: no upscattering iterations are performed, only one inner iteration completed per outer, using source iteration with the previous outer iterate as the initial guess.;In the Newton approach an extra degree of freedom is introduced by including the eigenvalue as an unknown, meaning an additional relation is necessary to close the system. In the diffusion theory case a normalization condition on the eigenvector was generally used, however in transport theory a number of so-called constraint relations were considered. These fall into two categories: normalization relations and eigenvalue update formulations. It was observed that the most effective of these constraint relations is the fission-rate eigenvalue update, derived directly from the eigenvalue update formula traditionally used to solve the k-eigenvalue problem.;Numerical results, including measured performance quantified in number of iterations and execution time, were generated for suites of benchmark problems using the various Newton's Method formulations for the k-eigenvalue problem in both transport and diffusion theories. These results showed that the choice of the perturbation parameter in the JFNK approximation has very little impact on the calculation while the choice of GMRES stopping criterion significantly affects the total cost of the calculation.;Overall, the numerical results showed that the Newton formulation of the k-eigenvalue problem in diffusion theory is competitive with the Chebyshev accelerated power iteration, with the JFNK(PI) formulation generally resulting in quicker execution times. The transport results showed that a number of the Newton formulations developed result in methods that are significantly less computationally expensive than traditional techniques. Results for the well-known C5G7-MOX benchmark problem demonstrate that the Newton approach reduces by a factor of 5 the total number of sweeps necessary to converge the point-wise fission source error to 10-4. (Abstract shortened by UMI.)

### Product Details

• ISBN-13: 9781109660791
• Publisher: ProQuest LLC
• Sold by: Barnes & Noble
• Format: eTextbook
• Pages: 289
• File size: 3 MB

## Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

(0)

(0)

(0)

(0)

### 1 Star

(0)

Your Name: Create a Pen Name or

### Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

### Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

### What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

### Reviews should not contain any of the following:

• - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
• - Time-sensitive information such as tour dates, signings, lectures, etc.
• - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
• - Comments focusing on the author or that may ruin the ending for others
• - Phone numbers, addresses, URLs
• - Pricing and availability information or alternative ordering information

### Reminder:

• - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
• - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
Search for Products You'd Like to Recommend

### Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

If you find inappropriate content, please report it to Barnes & Noble
Why is this product inappropriate?