Alternating Direction Implicit Schemes for the Two-Dimensional Time Fractional Nonlinear Super-Diffusion Equations

Abstract

As is known, there exist numerous alternating direction implicit (ADI) schemes for the two-dimensional linear time fractional partial differential equations (PDEs). However, if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems, the stability and convergence of the methods are often not clear. In this paper, two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integro-differential equations. In these two schemes, the standard second-order central difference approximation is used for the spatial discretization, and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time. The solvability, unconditional stability and $L_2$ norm convergence of the proposed ADI schemes are proved rigorously. The convergence order of the schemes is $O(τ + h^2_x + h^2_y)$, where $τ$ is the temporal mesh size, $h_x$ and $h_y$ are spatial mesh sizes in the $x$ and $y$ directions, respectively. Finally, numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.