ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation

In this paper, a two-dimensional fractional advection-dispersion equation (2D-FADE) with variable coefficients on a finite domain is considered. We use a new technique of combination of the Alternating Directions Implicit-Euler method (ADI-Euler), the unshifted Grünwald formula for the advection term, the right-shifted Grünwald formula for the diffusion term, and a Richardson extrapolation to establish an unconditionally stable second order accurate difference method. Stability, consistency and convergence of the ADI-Euler method for 2D-FADE are examined. A numerical example with known exact solution is also presented, and the behavior of the error is analyzed to verify the order of convergence of the ADI-Euler method and the extrapolated ADI-Euler method. © 2007 KSCAM and Springer-Verlag.