Jacobian-free Newton–Krylov methods for turbomachinery steady aerodynamic analysis

Abstract

This paper presents the algorithm development of a Jacobian-free Newton–Krylov (JFNK) solver for turbomachinery steady aerodynamic analysis, based on the three-dimensional Favre-averaged Navier–Stokes (FANS) equations. The Jacobian-free approach achieves accurate Jacobian-vector calculation using automatic differentiation, avoiding the complexity of explicitly forming a full Jacobian matrix. A lower–upper Symmetric-Gauss–Seidel (LU-SGS) preconditioner and a geometric multigrid preconditioner are proposed in this work to further enhance convergence. The two preconditioners are also compared with the standard incomplete LU factorization (ILU) preconditioner. The LU-SGS preconditioner does not rely on any explicit matrix, resulting in a fully matrix-free Newton–Krylov solver, while both the ILU and multigrid preconditioners require the explicit formation of an approximate Jacobian matrix. The performance of the developed JFNK solver is benchmarked against that of a standalone multigrid solver, which employs a multistage Runge–Kutta time-stepping scheme with an LU-SGS residual smoother. Two transonic compressor test cases are used for performance evaluation. Numerical results consistently reveal that the multigrid-preconditioned JFNK solver achieves a faster convergence rate than the standalone multigrid solver, while the other preconditioning cases require significantly more computational time. These findings demonstrate the effectiveness and robustness of the proposed multigrid preconditioner.