Physics‐based preconditioning of Jacobian‐free Newton–Krylov solver for Navier–Stokes equations using nodal integral method

Abstract

The nodal integral methods (NIMs) have found widespread use in the nuclear industry for neutron transport problems due to their high accuracy. However, despite considerable development, these methods have limited acceptability among the fluid flow community. One major drawback of these methods is the lack of robust and efficient nonlinear solvers for the algebraic equations resulting from discretization. Since its inception, several modifications have been made to make NIMs more agile, efficient, and accurate. Modified nodal integral method (MNIM) and modified MNIM (M2NIM) are the two most recent and efficient versions of the NIM for fluid flow problems. M2NIM modifies the MNIM by replacing the current time convective velocity with the previous time convective velocity, leading to faster convergence albeit with reduced accuracy. This work proposes a preconditioned Jacobian‐free Newton–Krylov approach for solving the Navier–Stokes equation using MNIM. The Krylov solvers do not generally work well without an appropriate preconditioner. Therefore, M2NIM is used here as a preconditioner to accelerate the solution of MNIM. Due to pressure–velocity coupling in the Navier–Stokes equation, developing a quality preconditioner for these equations needs significant effort. The momentum equation is solved using the time‐splitting alternate direction implicit method. The velocities obtained from the solution are then used to solve the pressure Poisson equation. The computational results for the Navier–Stokes equation are presented to underscore the advantages of the developed algorithm.

An Efficient and Robust ILU(k) Preconditioner for Steady-State Neutron Diffusion Problem Based on MOOSE

“…Although it is still time-consuming, it can improve efficiency by reducing the number of iteration steps. As a result, The performance of ILU (11) with the ND algorithm is slightly superior to ILU(3) with natural ordering. So, although natural ordering only could achieve good computational efficiency at low fill-in levels, such as k = 3, the computational time will increase sharply as the fill-in level increases.…”

“…Compared with the traditional methods, such as the operator splitting method and Picard iteration method [4][5][6], the Jacobian-free Newton-Krylov (JFNK) algorithm [7] is a powerful solver for the nonlinear equations due to its high-order convergence rate. The JFNK algorithm has been widely used in the newly developed nuclear reactor simulator [8][9][10][11], such as the MOOSE platform [12] and LIME platform [13,14]. In detail, several nuclear engineering programs have been made by Idaho National Laboratory based on the MOOSE platform, such as the reactor physics code MAMOTH [15], the advanced thermal-hydraulic code Pronghorn [16], and the two-phase flow code RELAP-7 [17].…”