Over the past twenty years, the mathematical modeling of Fractional Differential Equations (FDEs) has attracted growing attention and expanded significantly across various scientific fields, including bioengineering, mathematical biology, physics, chemistry and computational medicine. In this context, finding analytical solutions of FDEs is often more challenging than for classical ordinary differential equations, while the derivation of accurate and reliable numerical methods suffers from the possible non-smoothness of the solution and/or the vector field at the starting time, not to mention that the efficient treatment of the persistent memory term can make long-time simulations computationally demanding, due to the non-locality of the operator. To mitigate the aforementioned issues, the class of Runge-Kutta type methods, named Fractional HBVMs (FHBVMs), has been recently introduced ([1, 2]) and applied to solve initial value problems of fractional differential equations, releasing a corresponding Matlab© software ([2]). Though an error analysis has already been given, a corresponding linear stability analysis was still lack- ing. For this purpose, this presentation aims at bridging such a gap, providing a linear stability analysis in the continuous and discrete settings and allowing for a mixed graded/uniform stepsize strategy, that gives rise to an updated version of the code ([3]). As confirmed by the numerical tests, the novel approach is mainly tailored for problems with non-smooth vector field at the starting time, whose solution is both non-smooth at the origin and oscillatory, to concurrently gain accuracy in reproducing the initial non-smooth behaviour of the solution, while maintaining efficiency over extended time periods. [1] Brugnano, L., Burrage, K., Burrage, P., Iavernaro, F. A spectrally accurate step-by-step method for the numerical solution of fractional differential equations. J. Sci. Comput. (2024) 99(2):48. https://doi.org/10.1007/s10915-024-02517-1. [2] Brugnano, L., Gurioli, G., Iavernaro, F. Solving FDE-IVPs by using Fractional HB- VMs: Some experiments with the fhbvm code. J. Comput. Methods Sci. Eng. (2025) https://doi.org/10.1177/14727978251321328. [3] Brugnano, L., Gurioli, G., Iavernaro, F., and Vikerpuur, M. Analysis and implementation of collo- cation methods for fractional differential equations (2025). arXiv preprint arXiv:2503.17719 (submitted for publication).