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

• Published in: International journal for numerical methods in fluids 2024-02, Vol.96 (2), p.138-160
• Main Authors: Ahmed, Nadeem, Singh, Suneet, Kumar, Niteen
• Format: Article
• Language: English
• Subjects: Acceptability; Accuracy; Algorithms; Aquatic reptiles; Fluid flow; Implicit methods; Jacobian‐free Newton–Krylov method; Mathematical analysis; Momentum; Momentum equation; Navier-Stokes equations; Navier‐Stokes equation; nodal integral method; Physics; physics‐based preconditioning; Poisson equation; Preconditioning; Solvers; time‐splitting alternating direction implicit method; Velocity; Velocity coupling
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5236

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. A novel physics‐based preconditioner of the Jacobian‐free Newton–Krylov approach is developed to solve the Navier–Stokes equation using the NIM. The proposed preconditioner leads to huge reduction in condition number by clustering of eigenvalues. Therefore, GMRES convergence improves which drastically reduces Krylov iterations. The reduction in Krylov iterations saves the CPU runtime substantially.