*Article* **Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers**

**Galina Muratova 1,\*,†,‡ , Tatiana Martynova 1,†,‡ , Evgeniya Andreeva 1,†, Vadim Bavin 1,† and Zeng-Qi Wang <sup>2</sup>**


Received: 29 November 2019; Accepted: 21 January 2020; Published: 4 February 2020

**Abstract:** Multigrid methods (MGMs) are used for discretized systems of partial differential equations (PDEs) which arise from finite difference approximation of the incompressible Navier–Stokes equations. After discretization and linearization of the equations, systems of linear algebraic equations (SLAEs) with a strongly non-Hermitian matrix appear. Hermitian/skew-Hermitian splitting (HSS) and skew-Hermitian triangular splitting (STS) methods are considered as smoothers in the MGM for solving the SLAE. Numerical results for an algebraic multigrid (AMG) method with HSS-based smoothers are presented.

**Keywords:** multigrid methods; Hermitian/skew-Hermitian splitting method; skew-Hermitian triangular splitting method; strongly non-Hermitian matrix
