We study a model of magnetization switching driven by a spin current: the magnetization reversal can be induced without applying an external magnetic fi eld. We first write our one dimensional model in an adimensionalized form, using a small parameter $\epsilon$. We then explain the various time and space scales involved in the studied phenomena. Taking into account these scales, we fi rst construct an appropriate numerical scheme, that allows us to recover numerically various results of physical experiments. We then perform a formal asymptotic study as $\epsilon$ tends to 0, using a multiscale approach and asymptotic expansions. We thus obtain approximate limit models that we compare with the original model via numerical simulations.