Jacobian-Free Newton-Krylov Nodal Expansion Methods with Physics-Based Preconditioner and Local Elimination for Three-Dimensional and Multigroup k-Eigenvalue Problems

Abstract

Motivated by the high accuracy and efficiency of nodal methods on the coarse meshes and the superlinear convergence and high efficiency of Jacobian-free Newton-Krylov (JFNK) methods for large-scale nonlinear problems, a new JFNK nodal expansion method (NEM) with the physics-based preconditioner and local elimination NEM_JFNK is successfully developed to solve three-dimensional (3D) and multigroup k-eigenvalue problems by combining and integrating the NEM discrete systems into the framework of JFNK methods. A local elimination technique of NEM_JFNK is developed to eliminate some intermediate variables, expansion coefficients, and transverse leakage terms through equivalent transformation as much as possible in order to reduce the computational cost and the number of final-solving variables and residual equations constructed in NEM_JFNK. Then efficient physics-based preconditioners are successfully developed by approximating the matrices of the diffusion and removal terms, transverse leakage terms using the three-adjacent-node quadratic fitting methods, and scatter source terms, which make full use of the traditional power iteration. In addition, the Eisenstat-Walker forcing terms are used in the developed NEM_JFNK method to adaptively choose the convergence criterion of linear Krylov iteration within each Newton iteration based on the Newton residuals and to improve computational efficiency further. Finally, the NEM_JFNK code is developed for 3D and multigroup k-eigenvalue problems in neutron diffusion calculations and the detailed study of convergence, computational cost, and efficiency is carried out for several 3D problems. Numerical results show that the developed NEM_JFNK methods have faster convergence speed and are more efficient than the traditional NEM using power iteration, and the speedup ratio is greater for the higher convergence criterion.

Keywords:

• Jacobian-free Newton-Krylov
• nodal expansion method
• physics-based preconditioner

Acknowledgment

This research is supported by the Chinese National S&T Major Project ZX06901 and the Chinese National Natural Science Foundation Project 11375099.