Numerical Stability Analysis of Implicit Solution Methods for Harmonic Balance Equations

Abstract This article investigates the numerical stability of various implicit solution methods for efficiently solving harmonic balance equations for turbomachinery unsteady flows. Those solution methods implicitly integrate the time spectral source term to enhance stability and accelerate convergence. They include the block Jacobi method (BJ), the Jacobi iteration method (JI), and their variants, i.e., the modified block Jacobi method (MBJ) and the modified Jacobi iteration (MJI) method. These implicit treatments are typically combined with the lower-upper symmetric Gauss–Seidel method (LU-SGS) as a preconditioner of the Runge–Kutta scheme. The von Neumann analysis is applied to evaluate the stability and damping properties of all these methods. The findings reveal that the LU-SGS/BJ and LU-SGS/MJI schemes allow larger Courant numbers, in the order of hundreds, significantly improving convergence speed, while the LU-SGS/MBJ and LU-SGS/JI schemes fail to stabilize the solution, resulting in a Courant number below 10 as the grid-reduced frequency increases. The influence of Jacobi iterations on stabilization is also investigated. It is found that the minimum allowable relaxation factor does not change monotonically with the number of Jacobi iterations. Typically 2–4 Jacobi iterations are suggested for stability and efficiency, while more than four offers no benefit. The stability analysis results are verified by solving the harmonic balance equation system for two cases: inviscid flow over a 2D bump with a pressure disturbance at the outlet and turbulent flow in a 3D transonic compressor stage.