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

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.

Keywords

• Caputo fractional derivative
• time-fractional diffusion equation
• finite-difference scheme
• Thomas algorithm
• parallel sweep method
• accelerated over-relaxation method