The non-linear space-fractional diffusion equation

By Alex Simmons, Queensland University of Technology

Mathematical models are a popular field of research owing to their ability to describe physical problems, interactions or processes within the natural sciences (such as physics, biology, chemistry, geology). While a set of equations that describe these physical phenomena can be formed; determining a solution is usually a significantly more laborious task, in many cases it is not even possible. To circumvent these challenges a field of mathematics exists known as computational mathematics or numerical analysis. Computational mathematicians utilise a combination of computers and mathematical techniques to generate approximate solutions to a particular mathematical model.

In this project we focus our attention on a generalisation of a classical diffusion equation, known as the non-linear space-fractional diffusion equation. In essence, the model considers what happens when particles that would, in standard models, undergo diffusion are instead subjected to anomalous diffusion. This generalisation allows us to model a wider range of physical problems. Some applications include biokinematics, porous media flow, heart physiology and, salt-water intrusion in coastal aquifers.

The shift from modelling standard diffusion to anomalous diffusion sparks the requirement for efficient computational techniques in order to generate approximate solutions within a reasonable time-frame. Traditional techniques can be utilised, however the time taken and the hardware requirements to solve these problems can be many thousands of times more intensive when contrasted to the standard diffusion problems.

Iterative methods are a recurring theme in computational mathematics owing to their ability to approximate solutions through recursive refinement. Some of these iterative methods are referred to as the class of Krylov subspace methods, which form the basis of the techniques used to solve our anomalous diffusion model. The most significant issue faced with using iterative methods, such as Krylov subspace methods, is a high amount of recursions or iterations required to establish accurate approximations.

The focus of our work was the development of a preconditioner that accelerates the convergence of Krylov subspace methods and in-turn significantly reduces the amount of iterations required to produce accurate solutions. Our preconditioner works by taking the most important information or system dynamics within our model while discarding the heavy amount of minor interactions and forming a close approximation of what we expect our model to look like. By producing this close approximation the Krylov subspace method that we use is significantly accelerated. We can further accelerate convergence by taking more information from the model, which is represented by the measure of bandwidth; though there is a limiting factor on this.

Time taken for a simulation of a particular fractional diffusion model with varying bandwidths to be approximated. Clearly demonstrates that with a preconditioner, the runtime required is significantly reduced.

In the above table we present results for two different sized problems. The first is where we determined a solution at 16384 points in space and the second was at 65536 points in space. It is clear that with a preconditioner of bandwidth 201 we significantly reduced the runtime for the first test to just under 30seconds. For the second test the improvement was even more dramatic, with a preconditioner bandwidth of 401 we reduced the runtime from about 10hours down to 3.5minutes, which is 165 times faster.

With the improvements that we have been able to make with our preconditioner the realm of being able to solve these challenging fractional models becomes ever more possible. In-turn, this will allow scientists and mathematicians to be able to model increasingly more complicated systems where traditional models have failed.

Alex Simmons was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.