Computational challenges for an efficient simulation of anomalous diffusion problems

Abstract

In recent years, many non-conventional statistical phenomena characterized by unusual scaling exponents have been observed experimentally. Among them the anomalous diffusion stands out for its major impact in a variety of scientific disciplines. Several modeling techniques have been proposed already to mathematically describe this problem. In this context, fractional calculus has proven highly valuable for a more accurate modeling of many real-life processes and phenomena, since it is specially suited for characterizing the long-time memory and spatial heterogeneity effects typically found in any anomalous diffusion problem. However, these more accurate models pose fundamental challenges under a computational point of view. In this talk we review briefly the different mathematical strategies used so far to describe mathematically this phenomenon, the associated computational challenges, and finally close presenting some preliminary solutions.