Stability and Complexity Analyses of Finite Difference Algorithms for the Time-Fractional Diffusion Equation

TL;DR

This paper analyzes the stability and complexity of finite difference algorithms for the time-fractional diffusion equation, providing bounds for accurate simulations and proposing an efficient adaptive time step method.

Contribution

It characterizes the stability of existing algorithms, analyzes their time complexity, and introduces an improved adaptive time step approach using a linked list.

Findings

Stability bounds for finite difference algorithms are established.

The linked list implementation improves computational efficiency.

Adaptive time stepping balances speed and accuracy.

Abstract

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.