An ADI Crank-Nicolson Orthogonal Spline Collocation Method for the   Two-Dimensional Fractional Diffusion-Wave Equation
  A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit (ADI) method based on the Crank-Nicolson method combined with the $L1$-approximation of the time Caputo derivative of order $\alpha\in(1,2)$. It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence. 
Graeme Fairweather, Xuehua Yang, Da Xu and Haixiang Zhang
math.NA