Adaptive gauge method for long-time double-null simulations of spherical black-hole spacetimes

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.