Accelerating the iterative solution of convection-diffusion problems using singular value decomposition

TL;DR

This paper introduces a novel recycling strategy using singular value decomposition to accelerate the iterative solution of convection-diffusion problems, improving convergence in high-order numerical methods.

Contribution

It proposes a new SVD-based recycling approach for Krylov methods, enhancing convergence speed over traditional Arnoldi vector-based techniques.

Findings

SVD-based recycling improves convergence in convection-diffusion problems

The method is effective for high-order discretizations

Numerical tests show promising results

Abstract

The discretization of convection-diffusion equations by implicit or semi-implicit methods leads to a sequence of linear systems usually solved by iterative linear solvers such as GMRES. Many techniques bearing the name of \emph{recycling Krylov space methods} have been proposed to speed up the convergence rate after restarting, usually based on the selection and retention of some Arnoldi vectors. After providing a unified framework for the description of a broad class of recycling methods and preconditioners, we propose an alternative recycling strategy based on a singular value decomposition selection of previous solutions, and exploit this information in classical and new augmentation and deflation methods. The numerical tests in scalar non-linear convection-diffusion problems are promising for high-order methods.