A Multigrid Solver for Subdiffusion Equations

Articles in Press

Document Type : Research Paper

Authors

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.

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

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 07 May 2025

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History

  • Receive Date: 18 January 2025
  • Revise Date: 30 April 2025
  • Accept Date: 05 May 2025

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