Symmetry and separability of the neutron diffusion equation

Ramsey, Scott D.; Tellez, Jacob A.; Riewski, Eric J.; Temple, Brian A.

doi:10.1088/2399-6528/aae2a4

Title: Symmetry and separability of the neutron diffusion equation

Abstract

Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.

Authors:

^[1]; Tellez, Jacob A. ^[2]; Riewski, Eric J. ^[3]; Temple, Brian A. ^[1]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Univ. of Illinois, Urbana, IL (United States). Dept. of Nuclear, Plasma, and Radiological Engineering
Univ. of Wisconsin, Madison, WI (United States). Dept. of Engineering Physics

Publication Date:: Wed Sep 19 00:00:00 EDT 2018

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1482936

Report Number(s):: LA-UR-18-21934
Journal ID: ISSN 2399-6528

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Physics Communications

Additional Journal Information:: Journal Volume: 2; Journal Issue: 10; Journal ID: ISSN 2399-6528

Publisher:: IOP Publishing

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; neutron diffusion; symmetry analysis; separation of variables

Citation Formats


                    Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., and Temple, Brian A. Symmetry and separability of the neutron diffusion equation.  United States: N. p., 2018. 
Web.  doi:10.1088/2399-6528/aae2a4.

Copy to clipboard


                    Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., & Temple, Brian A. Symmetry and separability of the neutron diffusion equation.  United States.  https://doi.org/10.1088/2399-6528/aae2a4

Copy to clipboard


                    Ramsey, Scott D., Tellez, Jacob A., Riewski, Eric J., and Temple, Brian A. Wed .  
"Symmetry and separability of the neutron diffusion equation".  United States.  https://doi.org/10.1088/2399-6528/aae2a4.  https://www.osti.gov/servlets/purl/1482936.

Copy to clipboard


                    
@article{osti_1482936,

  title        = {Symmetry and separability of the neutron diffusion equation},

  author       = {Ramsey, Scott D. and Tellez, Jacob A. and Riewski, Eric J. and Temple, Brian A.},

  abstractNote = {Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.},

  doi          = {10.1088/2399-6528/aae2a4},

  journal      = {Journal of Physics Communications},

  number       = 10,

  volume       = 2,

  place        = {United States},

  year         = {Wed Sep 19 00:00:00 EDT 2018},

  month        = {Wed Sep 19 00:00:00 EDT 2018}

}

Copy to clipboard

Journal Article:

Free Publicly Available Full Text

Accepted Manuscript (DOE)

Publisher's Version of Record

https://doi.org/10.1088/2399-6528/aae2a4

Other availability

Search WorldCat to find libraries that may hold this journal

Figures / Tables:

Figure A1.: Plot of the normalized scalar flux resulting from the space-time separable symmetry solution where Δk_∞ is constant, $γ$=ωr and $τ$=$α$t.

All figures and tables (2 total)

Save / Share:

Export Metadata

Save to My Library

Works referenced in this record:

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media
journal, January 2014

Tsyfra, Ivan; Czyżycki, Tomasz
Abstract and Applied Analysis, Vol. 2014
DOI: 10.1155/2014/724238

Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations
journal, February 2016

Polyanin, Andrei D.; Zhurov, Alexei I.
Communications in Nonlinear Science and Numerical Simulation, Vol. 31, Issue 1-3
DOI: 10.1016/j.cnsns.2015.06.035

Separation of variables of a generalized porous medium equation with nonlinear source
journal, November 2002

Estévez, P. G.; Qu, Changzheng; Zhang, Shunli
Journal of Mathematical Analysis and Applications, Vol. 275, Issue 1
DOI: 10.1016/s0022-247x(02)00214-7

Geometric Approach to Invariance Groups and Solution of Partial Differential Systems
journal, April 1971

Harrison, B. Kent; Estabrook, Frank B.
Journal of Mathematical Physics, Vol. 12, Issue 4
DOI: 10.1063/1.1665631

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations
journal, August 1999

Chou, Kai-Seng; Qu, Changzheng
Journal of Physics A: Mathematical and General, Vol. 32, Issue 35
DOI: 10.1088/0305-4470/32/35/306

The Differential Form Method for Finding Symmetries
journal, August 2005

Harrison, B. Kent
Symmetry, Integrability and Geometry: Methods and Applications
DOI: 10.3842/sigma.2005.001

Figures / Tables found in this record:

Figure A1. (p. 15)

Figure B1. (p. 16)

Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.

Similar Records in DOE PAGES and OSTI.GOV collections:

Conditions for translation and scaling invariance of the neutron diffusion equation

Journal Article Giron, Jesse Frank ; Ramsey, Scott D. ; Temple, Brian Allen - Progress in Nuclear Energy

Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions ofmore »« less
Cited by 4
https://doi.org/10.1016/j.pnucene.2018.10.005

Full Text Available
Lie group invariant finite difference schemes for the neutron diffusion equation

Thesis/Dissertation Jaegers, Peter James

Finite difference techniques are used to solve a variety of differential equations. For the neutron diffusion equation, the typical local truncation error for standard finite difference approximation is on the order of the mesh spacing squared. To improve the accuracy of the finite difference approximation of the diffusion equation, the invariance properties of the original differential equation have been incorporated into the finite difference equations. Using the concept of an invariant difference operator, the invariant difference approximations of the multi-group neutron diffusion equation were determined in one-dimensional slab and two-dimensional Cartesian coordinates, for multiple region problems. These invariant difference equationsmore »« less
https://doi.org/10.2172/10165908

Full Text Available
Infinite-Order Symmetries for Quantum Separable Systems

Journal Article Miller, W ; Kalnins, E G ; Kress, J M ; ... - Physics of Atomic Nuclei

We develop a calculus to describe the (in general) infinite-order differential operator symmetries of a nonrelativistic Schroedinger eigenvalue equation that admits an orthogonal separation of variables in Riemannian n space. The infinite-order calculus exhibits structure not apparent when one studies only finite-order symmetries. The search for finite-order symmetries can then be reposed as one of looking for solutions of a coupled system of PDEs that are polynomial in certain parameters. Among the simple consequences of the calculus is that one can generate algorithmically a canonical basis for the space. Similarly, we can develop a calculus for conformal symmetries of themore »« less
https://doi.org/10.1134/1.2121926
Reduced Order Modeling for Prediction and Control of Large-Scale Systems.

Technical Report Kalashnikova, Irina ; Arunajatesan, Srinivasan ; Barone, Matthew Franklin ; ...

This report describes work performed from June 2012 through May 2014 as a part of a Sandia Early Career Laboratory Directed Research and Development (LDRD) project led by the first author. The objective of the project is to investigate methods for building stable and efficient proper orthogonal decomposition (POD)/Galerkin reduced order models (ROMs): models derived from a sequence of high-fidelity simulations but having a much lower computational cost. Since they are, by construction, small and fast, ROMs can enable real-time simulations of complex systems for onthe- spot analysis, control and decision-making in the presence of uncertainty. Of particular interest tomore »« less
https://doi.org/10.2172/1177206

Full Text Available
Similarity solution and Lie symmetry for a coupled nonlinear system

Journal Article Chowdhury, A R ; Ghose, A ; Naskar, M - Int. J. Theor. Phys.; (United States)

The Lie point symmetries of a set of coupled nonlinear partial differential equations are considered. The system is an extended version of the usual nonlinear Schroedinger equation. In the similarity variable deduced from the symmetry analysis, the system is equivalent to the Painleve III in Ince's classification. By starting from a solution of the Painleve equation, one can reproduce various classes of solutions of the original PDEs. Such solutions include both rational and progressive types or a combination of the two.
https://doi.org/10.1007/BF00672244

Similar Records

Title: Symmetry and separability of the neutron diffusion equation

Abstract

Citation Formats

Figures / Tables:

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media journal, January 2014

Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations journal, February 2016

Separation of variables of a generalized porous medium equation with nonlinear source journal, November 2002

Geometric Approach to Invariance Groups and Solution of Partial Differential Systems journal, April 1971

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations journal, August 1999

The Differential Form Method for Finding Symmetries journal, August 2005

Symmetry and Solution of Neutron Transport Equations in Nonhomogeneous Media
journal, January 2014

Direct functional separation of variables and new exact solutions to axisymmetric unsteady boundary-layer equations
journal, February 2016

Separation of variables of a generalized porous medium equation with nonlinear source
journal, November 2002

Geometric Approach to Invariance Groups and Solution of Partial Differential Systems
journal, April 1971

Symmetry groups and separation of variables of a class of nonlinear diffusion-convection equations
journal, August 1999

The Differential Form Method for Finding Symmetries
journal, August 2005