Global Error Bounds for the Petrov-Galerkin Discretization of the Neutron Transport Equation
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.
- Research Organization:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-48
- OSTI ID:
- 917492
- Report Number(s):
- UCRL-PROC-209167; TRN: US0805041
- 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
A Petrov-Galerkin finite element method for solving the neutron transport equation
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journal | May 1986 |
Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation
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journal | March 1970 |
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