Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation

Chang, B; Brown, P; Greenbaum, A; Machorro, E

doi:10.1002/nla.718

Title: Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation

Abstract

In this paper, we prove that the numerical solution of the mono-directional neutron transport equation by the Petrov-Galerkin method converges to the true solution in the L{sup 2} norm at the rate of h{sup 2}. Since consistency has been shown elsewhere, the focus here is on stability. We prove that the system of Petrov-Galerkin equations is stable by showing that the 2-norm of the inverse of the matrix for the system of equations is bounded by a number that is independent of the order of the matrix. This bound is equal to the length of the longest path that it takes a neutron to cross the domain in a straight line. A consequence of this bound is that the global error of the Petrov-Galerkin approximation is of the same order of h as the local truncation error. We use this result to explain the widely held observation that the solution of the Petrov-Galerkin method is second accurate for one class of problems, but is only first order accurate for another class of problems.

Authors:: Chang, B; Brown, P; Greenbaum, A; Machorro, E

Publication Date:: Fri Jan 21 00:00:00 EST 2005

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 917492

Report Number(s):: UCRL-PROC-209167
Journal ID: ISSN 1070-5325; TRN: US0805041

DOE Contract Number:: W-7405-ENG-48

Resource Type:: Conference

Resource Relation:: Journal Volume: 18; Journal Issue: 1; Conference: Presented at: Nuclear Explosive Code Design Conference, Livermore , CA, United States, Oct 04 - Oct 07, 2004

Country of Publication:: United States

Language:: English

Subject:: 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 22 GENERAL STUDIES OF NUCLEAR REACTORS; APPROXIMATIONS; DESIGN; GALERKIN-PETROV METHOD; NEUTRON TRANSPORT; NEUTRONS; NUCLEAR EXPLOSIVES; NUMERICAL SOLUTION; STABILITY

Citation Formats


                    Chang, B, Brown, P, Greenbaum, A, and Machorro, E. Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation.  United States: N. p., 2005. 
        Web.  doi:10.1002/nla.718.

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                    Chang, B, Brown, P, Greenbaum, A, & Machorro, E. Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation.  United States.  https://doi.org/10.1002/nla.718

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                    Chang, B, Brown, P, Greenbaum, A, and Machorro, E. 2005.  
        "Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation".  United States.  https://doi.org/10.1002/nla.718.  https://www.osti.gov/servlets/purl/917492.

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@article{osti_917492,

  title        = {Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation},

  author       = {Chang, B and Brown, P and Greenbaum, A and Machorro, E},

  abstractNote = {In this paper, we prove that the numerical solution of the mono-directional neutron transport equation by the Petrov-Galerkin method converges to the true solution in the L{sup 2} norm at the rate of h{sup 2}. Since consistency has been shown elsewhere, the focus here is on stability. We prove that the system of Petrov-Galerkin equations is stable by showing that the 2-norm of the inverse of the matrix for the system of equations is bounded by a number that is independent of the order of the matrix. This bound is equal to the length of the longest path that it takes a neutron to cross the domain in a straight line. A consequence of this bound is that the global error of the Petrov-Galerkin approximation is of the same order of h as the local truncation error. We use this result to explain the widely held observation that the solution of the Petrov-Galerkin method is second accurate for one class of problems, but is only first order accurate for another class of problems.},

  doi          = {10.1002/nla.718},

  url          = {https://www.osti.gov/biblio/917492},
  journal      = {},

  issn         = {1070-5325},

  number       = 1,

  volume       = 18,

  place        = {United States},

  year         = {Fri Jan 21 00:00:00 EST 2005},

  month        = {Fri Jan 21 00:00:00 EST 2005}

}

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Works referenced in this record:

A Petrov-Galerkin finite element method for solving the neutron transport equation
journal, May 1986

Greenbaum, A.; Ferguson, J. M.
Journal of Computational Physics, Vol. 64, Issue 1
https://doi.org/10.1016/0021-9991(86)90020-3

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation
journal, March 1970

Bramble, J. H.; Hilbert, S. R.
SIAM Journal on Numerical Analysis, Vol. 7, Issue 1
https://doi.org/10.1137/0707006

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Title: Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation

Abstract

Citation Formats

A Petrov-Galerkin finite element method for solving the neutron transport equation journal, May 1986

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation journal, March 1970

A Petrov-Galerkin finite element method for solving the neutron transport equation
journal, May 1986

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation
journal, March 1970