In this academic paper we present in detail the numerical solution of the accretion of a perfect fluid onto a black hole. The conditions are very simple, we consider a radial flux being accreted by a Schwarzschild black hole. We present two scenarios: 1) the test field case in which the fluid does not affect the geometry of the black hole space-time background, and 2) the full non-linear scenario, in which the geometry of the space-time evolves simultaneously with the fluid according to Einstein's equations. In the two scenarios we describe the black hole space-time in horizon penetrating coordinates, so that it is possible to visualize that accretion actually takes place within the numerical domain. For the evolution of matter we use the Valencia formulation of relativistic fluid dynamics. In the non-linear scenario we solve the equations of geometry using the ADM formulation of General Relativity, with very simple and intuitive gauge and boundary conditions, and include diagnostics related to the Apparent Horizon and Event Horizon growth. In view of the recent spectacular discoveries by the Event Horizon Telescope collaboration and further discoveries to come, the aim of this paper is to provide the necessary tools for interested graduate students in Black Hole Astrophysics, to enter into the accretion modeling starting from a considerable advanced starting point.