Simulating multi-dimensional anomalous diffusion in nonstationary media using variable-order vector fractional-derivative models with Kansa solver

Anomalous diffusion can be multiple dimensional and space dependent in large-scale natural media with evolving nonstationary heterogeneity, whose quantification requires an efficient technique. This research paper develops, evaluates, and applies variable-order, vector, spatial fractional-derivative equation (FDE) models with a Kansa solver, to capture spatiotemporal variation of super-diffusion along arbitrary angles (i.e., preferential pathways) in complex geological media. The Kansa approach is superior to the traditional Eulerian solvers in solving the vector FDE models, because it is meshless and can be conveniently extended to multi-dimensional transport processes. Numerical experiments show that the shape parameter, one critical parameter used in the Kansa solver, significantly affects the accuracy and convergence of the numerical solutions. In addition, the collocation nodes need to be assigned uniformly in the model domain to improve the numerical accuracy. Real-world applications also test the feasibility of this novel technique. Hence, the variable-order vector FDE model and the Kansa numerical solver developed in this study can provide a convenient tool to quantify complex anomalous transport in multi-dimensional and non-stationary media with continuously or abruptly changing heterogeneity, filling the knowledge gap in parsimonious non-local transport models developed in the last decades.