From a phenomenological point of view, the Gatenby-Gawlinski model of cancer invasion is inspired by the Warburg effect and relies on the acid-mediated invasion hypothesis. The main purpose consists in improving the understanding of its structural aspects, by means of a numerical approach based on finite volume approximations, and provide a space-averaged propagation speed estimate as a tool to investigate the phenomenon of traveling fronts. In particular, the multidimensional environment is explored with emphasis on some important qualitative aspects strongly linked to experimental observations.