Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

TL;DR

This paper introduces a new preconditioning approach using boundary value methods and GMRES for efficiently solving large linear systems arising from fractional diffusion equations, demonstrating rapid convergence and effectiveness through numerical experiments.

Contribution

It proposes a novel GMRES-based preconditioner with block circulant structure for boundary value method discretizations of fractional diffusion equations, ensuring fast convergence.

Findings

Preconditioned GMRES converges within a limited number of iterations.

The preconditioner is invertible and decomposes as I+L with bounded rank.

Numerical experiments confirm the method's efficiency and accuracy.

Abstract

The finite difference scheme with the shifted Gr\"{u}nwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed for solving these linear systems. One of the main results is that if an $A_{\nu_1,\nu_2}$-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I+L, where I is the identity matrix and the rank of L is at most $2m(\nu_1+\nu_2)$. It means that when the GMRES method is applied to solve the preconditioned linear systems, the method will converge in at most $2m(\nu_1+\nu_2)+1$ iterations.Finally, extensive numerical experiments are reported to illustrate the effectiveness of our methods for solving the fractional diffusion equations.