Finite difference Jacobian based Newton-Krylov coupling method for solving multi-physics nonlinear system of nuclear reactor

Abstract

Newton-Krylov (NK) method with the explicit Jacobian matrix is a promising method to solve the nonlinear multi-physics coupling system. However, how to form the Jacobian matrix efficiently is a key issue that is the most cumbersome step in NK method. In this work, a new solution with the finite difference Jacobian based NK (DJNK) method is proposed, where the Jacobian matrix is calculated by finite difference approximation instead of the analytical form. Moreover, DJNK is further improved to form Jacobian by calculating the diagonals of the submatrixes, leading to significant reduction of the time cost. Two benchmarks and a complicated transient neutronics/thermal-hydraulics coupling problem are conducted. Results show that DJNK outperforms widely used Jacobian-free NK (JFNK) method, because of the higher performance in matrix-vector product calculation and the more efficient preconditioning in Krylov iteration. A speedup ratio of DJNK over JFNK is observed to be 1.7.