Numerical methods and analysis for a class of fractional advection-dispersion models

In this paper, a class of fractional advection-dispersion models (FADMs) is considered. These models include five fractional advection-dispersion models, i.e., the times FADM, the mobile/immobile time FADM with a time Caputo fractional derivative 0 < γ < 1, the space FADM with two sides Riemann-Liouville derivatives, the timespace FADM and the time fractional advection-diffusion-wave model with damping with index 1 < γ < 2. These equations can be used to simulate the regional-scale anomalous dispersion with heavy tails. We propose computationally effective implicit numerical methods for these FADMs. The stability and convergence of the implicit numerical methods are analysed and compared systematically. Finally, some results are given to demonstrate the effectiveness of theoretical analysis.