A novel high-order ADI method for 3D fractionalconvection–diffusion equations

In this paper, a novel high-order alternating direction implicit (ADI) method is proposed for the three-dimensional (3D) fractional convection–diffusion equation with a temporal fractional-derivative α ∈ (0, 1). In order to keep the fourth order accuracy to approximate the second order derivatives and the desirable tridiagonal nature of the finite difference equations, we first propose a transformation to eliminate the convection terms. Then the Riemann–Liouville fractional integral operator is used to eliminate the temporal fractional-derivative. Finally, two ADI schemes with convergence order O(τmin(1+2α,2)+h4)Oτmin1+2α,2+h4 and O(τ2+h4)Oτ2+h4 are established respectively, where τ and h are the temporal and spatial step sizes. Numerical experiments are presented to show the high accuracy of the new method in comparison with the related works.