Numerical stability analysis of solution methods for steady and harmonic balance equations

This study investigates the stability and convergence properties of solution methods for the steady and unsteady Euler equations. A central scheme with artificial dissipation is used for spatial discretization. A Runge-Kutta scheme is used for the pseudo-time integration. The implicit residual smoothing for steady and harmonic balance solutions is achieved using the Lower-Upper Symmetric Gauss-Seidel(LU-SGS) method and the Lower-Upper Symmetric Gauss-Seidel/Block Jacobi(LU-SGS/BJ) method, respectively. Both the von Neumann and matrix methods are used to analyze the stability of the involved solution methods. The stability of these schemes obtained by the two stability analysis methods is identical for periodic boundary conditions as expected. However, only the matrix method can analyze the stability of solutions involving inlet, outlet, and slip wall boundary conditions. It is found that these boundary conditions enhance the stability of solution methods. Furthermore, the matrix method can also allow for the analysis of the impacts of non-uniformity in grids and flow fields on solution stability. For the harmonic balance equation system, the stability analysis demonstrates that the time spectral source term must be integrated implicitly to avoid instability for analysis with a large maximum grid-reduced frequency. The conclusions can be readily extended to the Reynolds averaged Navier-Stokes equations and verified using a Laval nozzle and the NASA rotor 37.