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

摘要

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.