  We implement a spatially fixed mesh refinement under spherical symmetry for
the characteristic formulation of General Relativity. The
Courant-Friedrich-Levy (CFL) condition lets us deploy an adaptive resolution in
(retarded-like) time, even for the nonlinear regime. As test cases, we
replicate the main features of the gravitational critical behavior and the
spacetime structure at null infinity using the Bondi mass and the News
function. Additionally, we obtain the global energy conservation for an extreme
situation, i.e. in the threshold of the black hole formation. In principle, the
calibrated code can be used in conjunction with an ADM 3+1 code to confirm the
critical behavior recently reported in the gravitational collapse of a massless
scalar field in an asymptotic anti-de Sitter spacetime. For the scenarios
studied, the fixed mesh refinement offers improved runtime and results
comparable to code without mesh refinement.
