This study looks at a governing equation of Lévy motion. This governing equation is a generalization of the classical, deterministic advection diffusion equation, just as Lévy motion is a generalization of Brownian motion. It can be represented by fractional derivatives, which are integrodifferential operators, describing a spatially nonlocal process. A high order explicit numerical method is presented for advection dominated problems in the presence of anomalous diffusion. The order of convergence of the numerical method varies between two and three and for advection dominated flows is close to three. Although the method is conditionally stable, the restrictions allow wide stability regions. Some numerical experiments will be presented to show the performance of the method and to observe the anomalous diffusion.