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Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. (English) Zbl 1251.65126
The authors introduce and investigate an alternating direction implicit (ADI) method for numerically solving the two-dimensional time-fractional wave equation with Caputo type derivatives. The method uses a high order approach for the space discretization. The authors provide a rigorous convergence and stability analysis. In this way, they demonstrate that the algorithm can obtain highly accurate results without requiring very fine spatial meshes. In view of the memory properties of the fractional differential operator with respect to the time variable, this feature leads to a substantial reduction of the memory and runtime requirements of the algorithm if the solution of the differential equation is smooth (an assumption that, however, will in practice frequently not be satisfied). A simpler Crank-Nicoloson type algorithm is also described and compared to the first method.
MSC:
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
35R11Fractional partial differential equations
35L05Wave equation (hyperbolic PDE)