A multigrid solver for subdiffusion equations

Mokhtari, Reza; Ramezani, Mohadeseh; Haase, Gundolf

doi:10.22034/cmde.2025.65520.3017

A multigrid solver for subdiffusion equations

Document Type : Research Paper

Authors

¹ 1. Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran. 2. Department for Mathematics and Scientific Computing, University of Graz, Graz, Austria.

² Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.

10.22034/cmde.2025.65520.3017

Abstract

In this paper, we investigate the S3-FD method for solving time-fractional diffusion equations in both one-dimensional (1D) and two-dimensional (2D) spatial domains, achieving high-order temporal accuracy. We leverage the S3 formula, which has a temporal accuracy of \(4 - \alpha\), to approximate the Caputo fractional derivative of order \(\alpha \in (0,1)\), and we employ the finite difference approach for spatial discretization. We develop a fully discrete scheme for both uniform and non-uniform spatial meshes. Our analysis begins with the 1D subdiffusion problem, where we employ the cyclic reduction method alongside OpenMP-based parallel programming to reduce computational costs. Leveraging this groundwork, we extend our technique to the 2D subdiffusion problem using a multigrid method and domain decomposition strategy paired with MPI programming. This innovative method yields an impressive temporal convergence order of \(\mathcal{O}(\Delta t^{4-\alpha})\). The performance and efficiency of the proposed S3-FD algorithm are demonstrated through numerical experiments, highlighting its potential for large-scale fractional diffusion problems.

Keywords