  The present numerical investigation of a Plasma Wakefield Acceleration
scenario in the weakly non linear regime with external injection is motivated
by the upcoming campaigns at the SPARC\_LAB test facility where the final goal
is to demonstrate modest gradient acceleration ($\sim$1 GV/m) with no quality
loss. The accelerated bunch can be envisioned to seed a free electron laser.
The numerical study has been conducted with the particle-in-cell code ${\tt
ALaDyn}$, an exhaustive description of the plasma-acceleration version is
provided. The configuration consider a two bunches setup with parameters in the
facility range, the bunches are generated and pre-accelerated up to 100 MeV by
a high brightness photo-injector prior plasma injection. To verify the working
point robustness we have considered case scenario where the driver bunch
reaches the plasma or with a larger dimension or with large emittance. We also
present an analytical approach based on the envelope equation that allows to
reduce the matching condition in the presence of a ramp. Here, we limit our
interest to a simplified theoretical case with a linear plasma ramp. As a final
aspect we propose to combine classical integrated bunch diagnostics with the
test by Shapiro-Wilk, a mathematical test to diagnose bunch deviation from a
Gaussian distribution.
