A fast Jacobian-Free Newton-Krylov iterative solver for eigenvalue search problems in the reactor physics

‎The Jacobian-Free Newton-Krylov (JFNK) method has been widely used in solving nonlinear equations arising in many applications‎. ‎In this paper‎, ‎the JFNK solver is examined as an alternative to the traditional power iteration method for calculation of the fundamental eigenmode in reactor analysis based on even-parity neutron transport theory‎. ‎Since the Jacobian is not formed the only extra storage required is associated with the workspace of the Krylov solver used at every Newton step‎. ‎A new nonlinear function is developed for the even-parity neutron transport equation utilized to solve the eigenvalue problem using the JFNK‎. ‎This Newton-based method is compared with the standard iterative power method for a number of multi-groups‎, ‎one and two dimensional neutron transport benchmarks‎. ‎The results show that the proposed algorithm generally ends with fewer iterations and shorter run times than those of the traditional power method‎.