The simulation of time-periodic unsteady flows is a central problem in aeronautical applications, especially in turbomachinery. The so-called harmonic balance method which uses a spectral discretisation of the time-derivative has been shown to be a highly efficient approach for applications in unsteady aerodynamics and nonlinear aeroelasticity [2, 1]. In contrast to linearised frequency-domain methods, harmonic balance takes the nonlinear interaction between harmonics into account. In this paper we discuss various solution methods for the harmonic balance equation systems. Our main focus is on implicit pseudotime marching methods and iterative linear solvers. It turns out that the choice of an appropriate method depends on the configuration and the degree of nonlinearity involved. We show that one can, in many situations, simplify the linear implicit systems, thereby reducing the computational costs considerably. The results, however, also indicate that such simplification come at the expense of robustness in off-design conditions (e.g. separated flow) in combination with high amplitude disturbances. This implies that an efficient overall simulation strategy amounts to applying simplifying assumptions where applicable, while the robust and computationally more costly variant should only be used for highly non-linear phenomena. References [1] C. Berthold, J. Gross, C. Frey, and M. Krack. \Development of a fully-coupled harmonic balance method and a refined energy method for the computation of flutterinduced Limit Cycle Oscillations of bladed disks with nonlinear friction contacts". In: Journal of Fluids and Structures 102 (2021). doi: 10.1016/j.jfluidstructs. 2021.103233. [2] K. C. Hall, J. P. Thomas, and W. S. Clark. \Computation of Unsteady Nonlinear Flows in Cascades Using a Harmonic Balance Technique". In: AIAA J. 40.5 (2002), pp. 879{886. doi: 10.2514/2.1754.