Alternating Direction Implicit Schemes for the Two-Dimensional Time Fractional Nonlinear Super-Diffusion Equations (Journal of Computational Mathematics, 2019)
As is known, there exist numerous alternating direction implicit (ADI) schemes for thetwo-dimensional linear time fractional partial differential equations (PDEs). However, ifthe ADI schemes for linear problems combined with local linearization techniques are appliedto solve nonlinear problems, the stability and convergence of the methods are oftennot clear. In this paper, two ADI schemes are developed for solving the two-dimensionaltime fractional nonlinear super-diffusion equations based on their equivalent partial integro-differentialequations. In these two schemes, the standard second-order central differenceapproximation is used for the spatial discretization, and the classical first-order approximationis applied to discretize the Riemann-Liouville fractional integral in time. Thesolvability, unconditional stability and $L_2$ norm convergence of the proposed ADI schemesare proved rigorously. The convergence order of the schemes is $O(τ + h^2_x + h^2_y)$, where $τ$ isthe 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 resultsand demonstrate the performances of two ADI schemes.
Keywords: Time fractional super-diffusion equation; Nonlinear system; ADI schemes; Stability; Convergence.