A Crank-Nicolson ADI quadratic spline collocation method for two-dimensional Riemann-Liouville space-fractional diffusion equations

In this paper, we develop a Crank-Nicolson ADI quadratic spline collocation method for the approximation of two-dimensional two-sided Riemann-Liouville space-fractional diffusion equation, in which a quadratic spline collocation method combined with ADI approach is considered for the discretization of the space-fractional derivatives with orders 1 < alpha, beta < 2, and a Crank-Nicolson method is proposed for the discretization of the first-order time derivative. The novel method is proved to be unconditionally stable for gamma(*) (approximate to 1.2576) < alpha, beta <= 2. Moreover, the method is shown to be convergent with second order in time and min{3 - alpha, 3 - beta} order in space, respectively. Finally, numerical examples are attached to confirm the theoretical results. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.