Advanced Computational Methods for Solving Multi-Scale Nonlinear Engineering Problems

Abstract

The classical integer-order transport equations, like the advection-diffusion equation, are not always useful in representing the anomalous transport process in complex environmental systems, especially in non-Fickian non-homogeneous porous media where long-range dependence and memory effects are predominant. To overcome these pitfalls, this paper introduces a fractional-order differentiation modeling system in order to implement non-local temporal and spatial dynamics. This is achieved by developing a generalized form of the transport model based on Caputo time-fractional and Riesz space-fractional derivatives in order to describe sub-diffusive and super-diffusive processes. It is the proposed model, which is numerically solved by a Grünwald-Letnikov-based discretization scheme (with a stable finite difference approach). The results of the simulations show that the fractional model is much better in predicting the concentration profiles than the classical models, and is effective in capturing the heavy tails and late transport behavior. The parametric analysis shows how the dynamics of a system are affected by the choice of the fractional orders, which can be used to understand the mechanisms of transport more thoroughly. The suggested framework provides a strong and physically uniform device to model anomalous transport used in the environmental engineering contexts such as groundwater contamination and pollutant dispersion.