  We investigate the RG-time integration of the effective potential in the
functional renormalization group in the presence of spontaneous symmetry
breaking and its subsequent convexity restoration on the example of a scalar
theory in $d=3$. The features of this setup are common to many physical models
and our results are, therefore, directly applicable to a variety of situations.
We provide exhaustive work-precision benchmarks and numerical stability
analyses by considering the combination of different discrete formulations of
the flow equation and a large collection of different algorithms. The results
are explained by using the different components entering the RG-time
integration process and the eigenvalue structure of the discrete system.
Particularly, the combination of Rosenbrock methods, implicit multistep methods
or certain (diagonally) implicit Runge-Kutta methods with exact or autodiff
Jacobians proves to be very potent. Furthermore, a reformulation in a
logarithmic variable circumvents issues related to the singularity bound in the
flat regime of the potential.
