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Using numerical techniques, we study the collapse of a scalar fieldconfiguration in the Newtonian limit of the spherically symmetricEinstein--Klein--Gordon (EKG) system, which results in the so calledSchr\"odinger--Newton (SN) set of equations. We present the numerical codedeveloped to evolve the SN system and topics related, like equilibriumconfigurations and boundary conditions. Also, we analyze the evolution ofdifferent initial configurations and the physical quantities associated tothem. In particular, we readdress the issue of the gravitational coolingmechanism for Newtonian systems and find that all systems settle down onto a0--node equilibrium configuration.Comment: RevTex file, 19 pages, 26 eps figures. Minor changes, matches version  to appear in PR

Evolution of the Schrödinger-Newton system for a self-gravitating scalar field