HSS(0): an Improved Hermitian/Skew-Hermitian Splitting Iteration

TL;DR

HSS(0) is an improved iterative method for solving non-Hermitian linear systems, offering faster convergence and less parameter sensitivity by modifying the Hermitian/skew-Hermitian splitting approach.

Contribution

The paper introduces HSS(0), a novel variant of the HSS method that enhances convergence speed and robustness through an analytical parameter optimization and modified iteration scheme.

Findings

HSS(0) converges faster than standard HSS.

HSS(0) is less sensitive to parameter choices.

Numerical tests confirm high efficiency on convection-diffusion problems.

Abstract

We propose an improved version of the Hermitian/skew-Hermitian splitting (HSS) iterative method, which we call HSS(0), to solve non-Hermitian linear systems with a positive definite Hermitian part. The improvement is based on solving the Hermitian half iteration without a shift, and applying a shift only for the skew-Hermitian solve. An optimal parameter is derived analytically, and a corresponding upper bound on the convergence speed is obtained. Using a combination of analytical proofs and numerical validations, we show that HSS(0) yields a dramatically faster convergence speed than standard HSS. Furthermore, HSS(0) is much less sensitive to the choice of the parameter. Numerical experiments on a convection-diffusion model problem in two and three dimensions illustrate the high efficiency of HSS(0).