New Splitting Iterative Methods for Solving Multidimensional Neutron Transport Equations
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This thesis focuses on iterative methods for the treatment of the steady state neutron transport equation in slab geometry, bounded convex domain of Rn (n = 2,3) and in 1D spherical geometry. We introduce a generic Alternate Direction Implicit (ADI)like iterative method based on positive definite and maccretive splitting (PAS) for linear operator equations with operators admitting such splitting. This method converges unconditionally and its SOR acceleration yields convergence results similar to those obtained in presence of finite dimensional systems with matrices possessing the Young property A. The proposed methods are illustrated by a numerical example in which an integrodifferential problem of transport theory is considered. In the particular case where the positive definite part of the linear equation operator is selfadjoint, an upper bound for the contraction factor of the iterative method, which depends solely on the spectrum of the selfadjoint part is derived. As such, this method has been successfully applied to the neutron transport equation in slab and 2D cartesian geometry and in 1D spherical geometry. The selfadjoint and maccretive splitting leads to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of minimal residual and preconditioned minimal residual algorithms using GaussSeidel, symmetric GaussSeidel and polynomial preconditioning are then applied to solve the matrix operator equation. Theoretical analysis shows that the methods converge unconditionally and upper bounds of the rate of residual decreasing which depend solely on the spectrum of the selfadjoint part of the operator are derived. The convergence of theses solvers is illustrated numerically on a sample neutron transport problem in 2D geometry. Various test cases, including pure scattering and optically thick domains are considered.
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ISBNs  9781599423968, 9781612333779, 1599423960, 9781612333779 
Language  English 
Number of Pages  159 