  Fractional differential equations (FDEs) are an extension of the theory of
fractional calculus. However, due to the difficulty in finding analytical
solutions, there have not been extensive applications of FDEs until recent
decades. With the advent of increasing computational capacity along with
advances in numerical methods, there has been increased interest in using FDEs
to represent complex physical processes, where dynamics may not be as
accurately captured with classical differential equations. The time-fractional
diffusion equation is an FDE that represents the underlying physical mechanism
of anomalous diffusion. But finding tractable analytical solutions to FDEs is
often much more involved than solving for the solutions of integer order
differential equations, and in many cases it is not possible to frame solutions
in a closed form expression that can be easily simulated or visually
represented. Therefore the development of numerical methods is vital. In
previous work we implemented the full 2D time-fractional diffusion equation as
a Forward Time Central Space finite difference equation by using the
Gr\"{u}nwald-Letnikov definition of the fractional derivative. In addition, we
derived an adaptive time step version that improves on calculation speed, with
some tradeoff in accuracy. Here, we explore and characterize stability of these
algorithms, in order to define bounds on computational parameters that are
crucial for performing accurate simulations. We also analyze the time
complexity of the algorithms, and describe an alternate adaptive time step
approach that utilizes a linked list implementation, which yields better
algorithmic efficiency.

Numerical Analysis;Numerical Analysis