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Error Analysis of the Nodal Integral Method for Solving the Neutron Diffusion Equation in Two-Dimensional Cartesian Geometry

R. M. Ferrer, Y. Y. Azmy

Nuclear Science and Engineering / Volume 162 / Number 3 / Pages 215-233

July 2009


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An error analysis is performed for the nodal integral method (NIM) applied to the one-speed, steady-state neutron diffusion equation in two-dimensional Cartesian geometry. The geometric configuration of the problem employed in the analysis consists of a homogeneous-material unit square with Dirichlet boundary conditions on all four sides. The NIM equations comprise three sets of equations: (a) one neutron balance equation per computational cell, (b) one current continuity condition per internal x = const computational cell edge, and (c) one current continuity condition per internal y = const computational cell edge. A Maximum Principle is proved for the solution of the NIM equations, followed by an error analysis achieved by applying the Maximum Principle to a carefully constructed mesh function driven by the truncation error or residual. The error analysis establishes the convergence of the NIM solution to the exact solution if the latter is twice differentiable. Furthermore, if the exact solution is four times differentiable, the NIM solution error is bounded by an O(a2) expression involving bounds on the exact solution's fourth partial derivatives, where a is half the scaled length of a computational cell. Numerical experiments are presented whose results successfully verify the conclusions of the error analysis.

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