A collocation method is developed for the solution of the one-dimensional neutron transport equation in slab geometry with both symmetric and polarly asymmetric scattering. In order to provide a method based on a standard approach to compare with the collocation method, the traditional discreteordinates method is extended to treat asymmetric scattering. For the symmetric scattering case, it is found that the collocation method offers a combination of some of the best characteristics of the

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... e-element and discrete-ordinates methods. For the asymmetric scattering case, it is found that the computational cost of cross-section data processing under the collocation approach can be significantly less than that associated with the discrete-ordinates approach. A general diffusion equation treating both symmetric and asymmetric scattering is developed and used in a synthetic acceleration algorithm to accelerate the iterative convergence of collocation solutions. It is shown that a certain type of asymmetric scattering can radically alter the asymptotic behavior of the transport solution and is mathematically equivalent within the diffusion approximation to particle transport under the influence of an electric field. The method easily extends to other geometries and higher dimensions. Applications exist in the areas of neutron transport with highly anisotropic scattering (such as that associatec with hydrogenous media), charged particle transport, and particle transport in controlled-fusion plasmas.