TY - JOUR
T1 - Alternating Direction Implicit Schemes for the Two-Dimensional Time Fractional Nonlinear Super-Diffusion Equations
AU - Huang , Jianfei
AU - Zhao , Yue
AU - Arshad , Sadia
AU - Li , Kuangying
AU - Tang , Yifa
JO - Journal of Computational Mathematics
VL - 3
SP - 297
EP - 315
PY - 2018
DA - 2018/09
SN - 37
DO - http://doi.org/10.4208/jcm.1802-m2017-0196
UR - https://global-sci.org/intro/article_detail/jcm/12723.html
KW - Time fractional super-diffusion equation, Nonlinear system, ADI schemes,
Stability, Convergence.
AB - <p style="text-align: justify;">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.</p>