A parallel algorithm for space-time-fractional partial differential equations

E Lorin

doi:10.1186/s13662-020-02744-4

Abstract

This paper is dedicated to the derivation of a simple parallel in space and time algorithm for space and time fractional evolution partial differential equations. We report the stability, the order of the method and provide some illustrating numerical experiments.

Highlights

1 Introduction This paper is devoted to the derivation of a parallel algorithm for solving fractional in space and time fractional partial differential equations
We will first focus on the parallelization in-time using a parareal algorithm, which is the most standard parallel-in-time algorithm for solving differential equations [20, 29]
We focus in the first part of the paper on the derivation and analysis of a parallel-in-time algorithm for fractional ordinary differential equations (FODEs) [40]

Summary

Introduction

This paper is devoted to the derivation of a parallel algorithm for solving fractional in space and time fractional partial differential equations. The parareal method allows for an accurate parallel computation of ODEs, and we refer to [16, 21, 29] for details about this celebrated method for solving ordinary/partial differential equations It is in particular successfully combined with traditional domain decomposition methods in space. Very few works exist on parallel-in-time methods for FODEs; let us cite [38] where a parareal method along with collocation and Fourier-based FODE solvers is developed At this stage, we do not consider realist models from the literature, but focus on toy-scalar linear equations, for which it is possible to provide a relatively precise analysis and to exhibit the strong convergence and efficiency properties.

Parareal-Gorenflo algorithm for fractional differential equations

Computational complexity

Space and time parallelization

Conclusion