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

Abstract 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 α , β 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 γ ⁎ ( ≈ 1.2576 ) α , β ≤ 2 . Moreover, the method is shown to be convergent with second order in time and min ⁡ { 3 − α , 3 − β } order in space, respectively. Finally, numerical examples are attached to confirm the theoretical results.