Convergence Acceleration of the Harmonic Balance Method using a Time-Level Preconditioner

The Harmonic Balance method is nowadays widely applied for numerically solving problems that are known to possess time-periodic solutions. Key reasons for its success are its wide range of applicability, relative ease of implementation, and computational efficiency compared to time-accurate approaches. The computational efficiency of the Harmonic Balance method is partly derived from the fact that it searches directly for a periodic solution, instead of integrating the governing equations in time until a periodic solution is reached. Convergence acceleration techniques such as multigrid, implicit residual smoothing and local time stepping may also be used to improve the efficiency of the Harmonic Balance method. This paper considers another option for accelerating convergence, namely a novel time-level preconditioner that can be applied to the Harmonic Balance residual locally in each computational cell. This preconditioner is derived from a rigorous stability analysis of the Harmonic Balance equations and is shown to give a speed-up factor of 2 when applied to simulations of laminar vortex shedding behind a circular cylinder.