Compact Alternating Direction Implicit Scheme for the Two-Dimensional Fractional Diffusion-Wave Equation

Abstract

In this paper, we consider the numerical method for solving the two-dimensional fractional diffusion-wave equation with a time fractional derivative of order $\alpha$ ($1<\alpha<2$). A difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping is proposed and analyzed. The unconditional stability and $H^1$ norm convergence of the scheme are proved rigorously. The convergence order is $\mathcal {O}(\tau^{3-\alpha}+h_1^4+h^4_2),$ where $\tau$ is the temporal grid size and $h_1,h_2$ are spatial grid sizes in the $x$ and $y$ directions, respectively. In addition, a Crank--Nicolson ADI scheme is presented and the corresponding error estimates are also established. Numerical results are presented to support our theoretical analysis and indicate that the compact ADI scheme reduces the storage requirement and CPU time successfully.