  Perturbations of Schwarzschild-Droste black holes in the Regge-Wheeler gauge
benefit from the availability of a wave equation and from the gauge invariance
of the wave function, but lack smoothness. Nevertheless, the even perturbations
belong to the C\textsuperscript{0} continuity class, if the wave function and
its derivatives satisfy specific conditions on the discontinuities, known as
jump conditions, at the particle position. These conditions suggest a new way
for dealing with finite element integration in time domain. The forward time
value in the upper node of the $(t, r^*$) grid cell is obtained by the linear
combination of the three preceding node values and of analytic expressions
based on the jump conditions. The numerical integration does not deal directly
with the source term, the associated singularities and the potential. This
amounts to an indirect integration of the wave equation. The known wave forms
at infinity are recovered and the wave function at the particle position is
shown. In this series of papers, the radial trajectory is dealt with first,
being this method of integration applicable to generic orbits of EMRI (Extreme
Mass Ratio Inspiral).
