Iterative schemes for obtaining dominant alpha-modes of the neutron diffusion equation

Two new methods of obtaining dominant prompt alpha-modes (sometimes referred to as time-eigenfunctions) of the multigroup neutron diffusion equation are discussed. In the first of these, we initially compute the dominant K-eigenfunctions and K-eigenvalues (denoted by λ1, λ2, λ3 … etc.; λ1 being equal to the Keff) for the given nuclear reactor model, by existing method based on sub-space iteration (SSI) which is an improved version of power iteration method. Subsequently, a uniformly distributed (positive or negative) 1/v absorber of sufficient concentration is added so as to make a particular eigenvalue λi equal to unity. This gives ith alpha-mode. This procedure is repeated to find all the required alpha-modes. In the second method, we solve the alpha-eigenvalue problem directly by SSI method. This is clearly possible for a sub-critical reactor for which the inverse of the dominant alpha-eigenvalues are also the largest in magnitude as required by the SSI method. Here, the procedure is made applicable even to a super-critical reactor by making the reactor model sub-critical by the addition of a 1/v   absorber. Results of these calculations for a 3-D two group PHWR test-case are given. These results are validated against the results as obtained by a completely different approach based on Orthomin(1) algorithm published earlier. The direct method based on the sub-space iteration strategy is found to be a simple and reliable method for obtaining any number of alpha-modes. Also comments have been made on the relationship between fundamental αα and k values.