A Scalable Framework for Solving Fractional Diffusion Equations

TL;DR

This paper introduces a scalable framework for solving fractional diffusion equations using eigenfunction expansion, emphasizing parallelization strategies and demonstrating high performance on large computing clusters.

Contribution

The paper presents a novel scalable approach employing eigenfunction expansion and parallelization for fractional diffusion equations on complex geometries.

Findings

Efficient parallel computation of eigenvalues and eigenvectors.

High performance demonstrated on the Frontera cluster.

Accurate solutions on simple geometries with known exact solutions.

Abstract

The study of fractional order differential operators is receiving renewed attention in many scientific fields. In order to accommodate researchers doing work in these areas, there is a need for highly scalable numerical methods for solving partial differential equations that involve fractional order operators on complex geometries. These operators have desirable special properties that also change the computational considerations in such a way that undermines traditional methods and makes certain other approaches more appealing. We have developed a scalable framework for solving fractional diffusion equations using one such method, specifically the method of eigenfunction expansion. In this paper, we will discuss the specific parallelization strategies used to efficiently compute the full set of eigenvalues and eigenvectors for a discretized Laplace eigenvalue problem and apply them to construct approximate solutions to our fractional order model problems. Additionally, we demonstrate the performance of the method on the Frontera computing cluster and the accuracy of the method on simple geometries using known exact solutions.