A Jacobian-free Newton-krylov Iterative Scheme for Criticality Calculations Based on the Neutron Diffusion Equation

Newton-Krylov methods, primarily using the Jacobian-Free Newton-Krylov (JFNK) approximation, are examined as an alternative to the traditional power iteration method for the calculation of the fundamental eigenmode in reactor analysis applications based on diffusion theory. One JFNK approach can be considered an acceleration technique for the standard power iteration as it is “wrapped around” the power method, allowing for simplified implementation in preexisting codes. 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. Another Newton-based method is developed which solves the generalized eigenvalue problem formulation of the k-eigenvalue problem, both using the JFNK approximation and utilizing the action of the true Jacobian matrix. These Newton-based methods were compared to the unaccelerated and Chebyshev-accelerated power method for a number of reactor models. Results show calculation of the fundamental mode using JFNK acceleration of the power method generally results in fewer iterations and shorter run times than when using the unaccelerated and Chebyshev-accelerated power methods. The Newton-based approaches implemented for the solution of the generalized eigenvalue problem formulation of the k-eigenvalue problem are shown to exhibit poor convergence properties for the iterations associated with the linearized Newton step, highlighting the need for an effective preconditioner.