A multigrid solver for subdiffusion equations

In this paper, we investigate the S۳-FD method for solving time-fractional diffusion equations in both one-dimensional (۱D) and two-dimensional (۲D) spatial domains, achieving high-order temporal accuracy. We leverage the S۳ formula, which has a temporal accuracy of \(۴ - \alpha\), to approximate the Caputo fractional derivative of order \(\alpha \in (۰,۱)\), 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 ۱D 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 ۲D subdiffusion problem using a multigrid method and domain decomposition strategy paired with MPI programming. This innovative method yields an impressive temporal convergence order of  \(\mathcal{O}(\Delta t^{۴-\alpha})\). The performance and efficiency of the proposed S۳-FD algorithm are demonstrated through numerical experiments, highlighting its potential for large-scale fractional diffusion problems.

کلیدواژه ها:

S۳ formula ، Subdiffusion equation ، multigrid method ، domain decomposition ، Parallel processing