A Matrix-Free Eigenvalue Solver for the Multigroup Neutron Diffusion Equation

摘要

The stationary neutron transport equation describes the neutron population and thus, the generated heat, inside a nuclear reactor core. Obtaining the solution of this equation requires to solve a generalized eigenvalue problem efficiently. The majority of the eigenvalue solvers use the factorization of the system matrices to construct preconditioners, such as the ILU decomposition or the ICC decomposition, to speed up the convergence of the methods. The storage of the involved matrices and incomplete factorization demands high quantities of computational memory although a the sparse format is used. This makes the computational memory the limiting factor for this kind of calculations in some personal computers. In this work, we propose a matrix-free preconditioned eigenvalue solver that does not need to have the.matrices allocated in memory explicitly. This method is based on the block inverse-free preconditioned Arnoldi method (BIFPAM) with the innovation that uses a preconditioner that is applied from matrix-vector operations. As well as reducing enormously the computational memory, this methodology removes the time to assembly the sparse matrices involved in the system. A two-dimensional and three-dimensional benchmarks are used to study the performance of the methodology proposed.