Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity a t = 0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order alpha (1 < alpha < 2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L-1 method (single-step) and Nonuniform Crank-Nicolson L1 - 2 method (multi-step). The L-1 method has order of convergence (OC) min(3 - alpha, gamma alpha) where gamma is the mesh grading parameter used in construction of the nonuniform mesh, and L1-2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD a t = 0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min(3 - alpha, gamma alpha) and second order convergence, respectively for all 1 < alpha < 2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t = 0.