  Double-null coordinates are highly useful in numerical simulations of
dynamical spherically-symmetric black holes (BHs). However, they become
problematic in long-time simulations: Along the event horizon, the truncation
error grows exponentially in the outgoing Eddington null coordinate - which we
denote $v_e$ - and runs out of control for a sufficiently long interval of
$v_e$. This problem, if not properly addressed, would destroy the numerics both
inside and outside the black hole at late times (i.e. large $v_e$). In this
paper we explore the origin of this problem, and propose a resolution based on
adaptive gauge for the ingoing null coordinate $u$. This resolves the problem
outside the BH - and also inside the BH, if the latter is uncharged. However,
in the case of a charged BH, an analogous large-$v_e$ numerical problem occurs
at the inner horizon. We thus generalize our adaptive-gauge method in order to
overcome the IH problem as well. This improved adaptive gauge, to which we
refer as the maximal-$\sigma$ gauge, allows long-$v$ double-null numerical
simulation across both the event horizon and the (outgoing) inner horizon, and
up to the vicinity of the spacelike $r=0$ singularity. We conclude by
presenting a few numerical results deep inside a perturbed charged BH, in the
vicinity of the contracting Cauchy Horizon.
