Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors

Murat A Sultanov,Yerkebulan Nurlanuly,Elena N Akimova,Vladimir E Misilov

doi:10.3390/math10030323

Abstract

The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time.

Highlights

The last decades have seen the significant rise of interest to the time-fractional differential equations [1,2,3,4,5]
We develop a parallel implementation of the algorithm for solving the time-fractional diffusion equation for the superscalar multicore processors using the OpenMP technology [33] and automatic vectorization by the Intel C++ Compiler.The parallelization is performed as follows
The algorithms are based on the finitedifference scheme for approximating the differential equation and the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving the systems of linear algebraic equations

Summary

Introduction

The last decades have seen the significant rise of interest to the time-fractional differential equations [1,2,3,4,5]. This is due to possibility to use these equations for modeling the multiple phenomena of anomalous diffusion and other processes with memory effects. Multiple experimental researches [6,7,8] showed that the assumption of the Brownian motion in the diffusion processes may not be sufficient for the accurate description of some physical processes. The fractional differential equations are the powerful mathematical tool for adequate description of many real physical processes, and their application field still grows. Development of the efficient numerical algorithms for solving the direct and inverse problems for fractional differential equations is of considerable theoretical and practical interest today

Methods

Results

Discussion

Conclusion