Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media

摘要

The fractal mobile/immobile model of the solute transport is based on the assumption that the waiting times in the immobile region follow a power-law, and this leads to the application of fractional time derivatives. The model covers a wide family of systems that include heat diffusion and ocean acoustic propagation. This paper develops an efficient computational technique, stemming from the radial basis function-generated finite difference (RBF-FD), to solve the fractal mobile-immobile transport model (FMTM). The time fractional derivative of the FMTM is discretized via the shifted Grünwald-Letnikov formula with second-order accuracy. On the other hand, the spatial derivative is approximated using the local RBF-FD method. The main benefit of the local collocation technique is that we only need to consider discretization points present in each of the sub-domains around the collocation point. The stability and convergence analysis of the proposed method are proven via the energy method in the L~2 space. The numerical results for the FMTM on regular and irregular domains confirm the theoretical formulation and efficiency of the proposed scheme.