In silico research in medicine is thought to reduce the need for expensive clinical trials under the condition of reliable mathematical models and accurate and efficient numerical methods. In the present work, we tackle the numerical simulation of reaction-diffusion equations modeling human ischemic stroke. This problem induces peculiar difficulties like potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of steep spatial gradients in the reaction fronts, spatially very localized. Furthermore, simulations on realistic 3D geometries are mandatory in order to describe correctly this type of phenomenon. We introduce a new resolution strategy based mainly on time operator splitting and a well conceived parallelization strategy, that takes into account complex geometries. We consider then a high order implicit time integration of the reaction and an explicit one for the diffusion term in order to build a time operator splitting scheme that exploits efficiently the special features of each problem. Thus, we aim at solving complete and realistic models including all time and space scales with conventional computing resources, that is on a reasonably powerful workstation. Consequently, 2D and also fully 3D numerical simulations of ischemic strokes for a realistic brain geometry, are conducted for the first time and shown to reproduce the dynamics observed on MR images in stroke patients. Beyond this major step, in order to improve accuracy and computational efficiency of the simulations, we indicate how the present numerical strategy can be coupled to spatial adaptive multiresolution schemes. Preliminary results in the framework of simple geometries allow to assess the proposed strategy for further developments.