Differential Quadrature and Cubature Methods for Steady-State Space-Fractional Advection-Diffusion Equations

Space-fractional advection-diffusion equation is a promising tool to describe the solute anomalous transport in underground water, and it has been extended to multi-dimensions with the help of weighted, fractional directional diffusion operator [Benson, Wheatcraft and Meerschaert (2000)]. Due to the nonlocal property of the space-fractional derivative, it is always a challenge to develop an efficient numerical solution method. The present paper extends the polynomial-based differential quadrature and cubature methods to the solution of steady-state spatial fractional advection-diffusion equations on a rectangular domain. An improved differential cubature method is proposed which accelerates the solution process considerably. Owing to the global interpolation nature these methods are more accurate and efficient than the finite element method. Numerical convergence is investigated thru one- and two- dimensional benchmark problems. The convergence can be improved after well-organized explicit formulas for weighting coefficients are obtained.