A second-order weighted ADI scheme with nonuniform time grids for the two-dimensional time-fractional telegraph equation

In this paper, based on the weighted alternating direction implicit method, we investigate a second-order scheme with variable steps for the two-dimensional time-fractional telegraph equation (TFTE). Firstly, we derive a coupled system of the original equation by the symmetric fractional-order reduction (SFOR) method. Then the renowned L2- \(1_\sigma \) $1_{\sigma }$ formula on graded meshes is employed to approximate the Caputo derivative and a weighted ADI scheme for the coupled problem is constructed. In addition, with the aid of the Grönwall inequality, the unconditional stability and convergence of the weighted ADI scheme are analyzed. Finally, the numerical experiments are shown to verify the effectiveness and correctness of theoretical results.