Numerical Upscaling of Subdiffusive Transport through Disordered Media with Finite Correlation Length

We develop a method which constructs, from random walk data collected in a disordered medium, a system of parabolic partial differential equations that can describe the underlying subdiffusive transport. This method is intended for cases where the scales of interest are comparable to the correlation length, and in which the medium is given through a computational procedure to generate material samples. Essentially, our approach is based on fitting the effective exponent over time of a specially crafted stochastic differential equation with Markovian switching to that of the random walk data. Since the master equation of the former is given exactly by said system of parabolic PDEs, we obtain an approximation of the transport described by the random walks. We include numerical experiments for the case of a percolation cluster slightly above criticality, in which a marked subdiffusive behavior can be observed, and show the effectiveness and robustness of our approach. [PUBLICATION ABSTRACT]