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Numerical Solution of the Linear Integral Boltzmann Equation

Dermott E. Cullen

Nuclear Science and Engineering

Volume 53 / Number 1 / January 1974 / Pages 93-106


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An iterative method is proposed for solving the homogeneous (i.e., critical) or inhomogeneous (i.e., source) linear integral Boltzmann equation for general geometry. By using successive approximations, these two classes of problems are shown to be mathematically equivalent. For the homogeneous problem, constraints on the algorithm regarding the existence of eigenvalues and the initial approximation are investigated. The algorithm is applied to isotropically scattering slabs and spheres and is compared to previously published results as well as to an independent extrapolation method., For the inhomogeneous problem, an improvement over the normal successive collision method via the use of a Neumann series expansion is used to allow economic parametric studies. Constraints on the algorithm and methods of efficiently terminating the infinite Neumann series are investigated. The solution via the proposed method as applied to isotropically scattering slabs and spheres is provided in a compact form for a range of multiplication factors and optical dimensions. The shape of the scalar flux distribution is explained., Extensions of the method to more complex problems are outlined; in particular, the solution to an energy-dependent problem in general geometry is obtained and the implications of the results are discussed.

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