Refined Algorithm for Solving High-Order Harmonics of Multi Group Neutron Diffusion Equation

In recent years, high order harmonic (or eigenvector) of neutron diffusion equation has been widely used in on-line monitoring system of reactor power. There are two kinds of calculation method to solve the equation: corrected power iteration method and Krylov subspace methods. Fu Li used the corrected power iteration method. When solving for the ith harmonic, it tries to eliminate the influence of the front harmonics using the orthogonality of the harmonic function. But its convergence speed depends on the occupation ratio. When the dominant ratios equal to 1 or close to 1, convergence speed of fixed source iteration method is slow or convergence can’t be achieved. Another method is the Krylov subspace method, the main idea of this method is to project the eigenvalue and eigenvector of large-scale matrix to a small one. Then we can solve the small matrix eigenvalue and eigenvector to get the large ones. In recent years, the restart Arnoldi method emerged as a development of Krylov subspace method. The method uses continuous reboot Arnoldi decomposition, limiting expanding subspace, and the orthogonality of the subspace is guaranteed using orthogonalization method. This paper studied the refined algorithms, a method based on the Krylov subspace method of solving eigenvalue problem for large sparse matrix of neutron diffusion equation. Two improvements have been made for a restarted Arnoldi method. One is that using an ingenious linear combination of the refined Ritz vector forms an initial vector and then generates a new Krylov subspace. Another is that retaining the refined Ritz vector in the new subspace, called, augmented Krylov subspace. This way retains useful information and makes the resulting algorithm converge faster. Several numerical examples are the new algorithm with the implicitly restart Arnoldi algorithm (IRA) and the implicitly restarted refined Arnoldi algorithm (IRRA). Numerical results confirm efficiency of the new algorithm.