The numerical solution of the transport equation, describing the fate of a passive scaler in a moving fluid, has been the object of intense research for the past few decades. So much interest concerning an apparently inoffensive (in many occasions it is even linear) equation may seem, at first sight, misplaced. However, there is a fundamental difficulty in the solution of the transport equation, which results from the fact that, while advection and dispersion are simultaneous processes, they promote mass transport very differently: in the case of advection, transport is along characteristic lines that follow the flow, while in the case of dispersion, it is both along and between characteristic lines.