  We develop a numerical solver, that extends the computational framework
considered in [Phys. Rev. D 65, 084016 (2002)], to include scalar perturbations
of nonrotating black holes. The nonlinear Einstein-Klein-Gordon equations for a
massless scalar field minimally coupled to gravity are solved in two spatial
dimensions (2D). The numerical procedure is based on the ingoing light cone
formulation for an axially and reflection symmetric spacetime. The solver is
second order accurate and was validated in different ways. We use for
calibration an auxiliary 1D solver with the same initial and boundary
conditions and the same evolution algorithm. We reproduce the quasinormal modes
for the massless scalar field harmonics $\ell = 0$, $1$ and $2$. For these same
harmonics, in the linear approximation, we calculate the balance of energy
between the black hole and the world tube. As an example of nonlinear harmonic
generation, we show the distortion of a marginally trapped two-surface
approximated as a q-boundary and based upon the harmonic $\ell=2$.
Additionally, we study the evolution of the $\ell = 8$ harmonic in order to
test the solver in a spacetime with a complex angular structure. Further
applications and extensions are briefly discussed.
