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

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## Abstract

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## 1. Introduction

## 2. Multigrid Methods

- Richardson’s Iterative method;
- Gauss–Jacobi method;
- Symmetric Gauss–Seidel method;
- Gauss–Seidel Alternate Direction method;
- Gauss–Seidel method with black and white ordering;
- Four-color Gauss–Seidel method;
- Iteration zebra method;
- Incomplete factorization method;
- Specially adapted SOR.

## 3. Smoothers Based on the HSS and the STS Iteration Methods

**The HSS iteration method**[1]: Given an initial guess ${v}^{\left(0\right)}$, for $k=0,1,2,\dots $ until $\left\{{v}^{\left(k\right)}\right\}$ convergence, compute

**The STS iteration method**[6,8]: Given an initial guess ${v}^{\left(0\right)}$ and two positive parameters $\omega $ and $\tau $. For $k=0,1,2,\dots $ until $\left\{{v}^{\left(k\right)}\right\}$ convergence, compute

## 4. Numerical Experiments

- Velocity field components ${u}^{\prime}={u}^{(n+1)}$ and ${v}^{\prime}={v}^{(n+1)}$ are determined by solving the implicit momentum equation with ${P}^{\prime}$, and for treating nonlinearity, the Newton linearization around the old time level is used.$$\frac{{u}_{ij}^{(n+1)}-{u}_{ij}^{\left(n\right)}}{\delta t}+\left({u}_{ij}^{\left(n\right)}\left(\frac{{u}_{ij}^{(n+1)}-{u}_{i-1,j}^{(n+1)}}{{h}_{1}}\right)+{u}_{ij}^{(n+1)}\left(\frac{{u}_{ij}^{\left(n\right)}-{u}_{i-1,j}^{\left(n\right)}}{{h}_{1}}\right)-{u}_{ij}^{\left(n\right)}\left(\frac{{u}_{ij}^{\left(n\right)}-{u}_{i-1,j}^{\left(n\right)}}{{h}_{1}}\right)\right)+$$$$+\left({v}_{ij}^{\left(n\right)}\left(\frac{{u}_{ij}^{(n+1)}-{u}_{i,j-1}^{(n+1)}}{{h}_{2}}\right)+{v}_{ij}^{(n+1)}\left(\frac{{u}_{ij}^{\left(n\right)}-{u}_{i,j-1}^{\left(n\right)}}{{h}_{2}}\right)-{v}_{ij}^{\left(n\right)}\left(\frac{{u}_{ij}^{\left(n\right)}-{u}_{i,j-1}^{\left(n\right)}}{{h}_{2}}\right)\right)-$$$$-{\displaystyle \frac{1}{Re}}\left(\frac{{u}_{i+1,j}^{(n+1)}-2{u}_{ij}^{(n+1)}+{u}_{i-1,j}^{(n+1)}}{{h}_{1}^{2}}+\frac{{u}_{i,j+1}^{(n+1)}-2{u}_{ij}^{(n+1)}+{u}_{i,j-1}^{(n+1)}}{{h}_{2}^{2}}\right)={f}_{1}^{\prime},$$$$\frac{{v}_{ij}^{(n+1)}-{v}_{ij}^{\left(n\right)}}{\delta t}+\left({u}_{ij}^{\left(n\right)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i-1,j}^{(n+1)}}{{h}_{1}}\right)+{u}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{\left(n\right)}-{v}_{i-1,j}^{\left(n\right)}}{{h}_{1}}\right)-{u}_{ij}^{\left(n\right)}\left(\frac{{v}_{ij}^{\left(n\right)}-{v}_{i-1,j}^{\left(n\right)}}{{h}_{1}}\right)\right)+$$$$+\left({v}_{ij}^{\left(n\right)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i,j-1}^{(n+1)}}{{h}_{2}}\right)+{v}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{\left(n\right)}-{v}_{i,j-1}^{\left(n\right)}}{{h}_{2}}\right)-{v}_{ij}^{\left(n\right)}\left(\frac{{v}_{ij}^{\left(n\right)}-{v}_{i,j-1}^{\left(n\right)}}{{h}_{2}}\right)\right)-$$$$-{\displaystyle \frac{1}{Re}}\left(\frac{{v}_{i+1,j}^{(n+1)}-2{v}_{ij}^{(n+1)}+{v}_{i-1,j}^{(n+1)}}{{h}_{1}^{2}}+\frac{{v}_{i,j+1}^{(n+1)}-2{v}_{ij}^{(n+1)}+{v}_{i,j-1}^{(n+1)}}{{h}_{2}^{2}}\right)={f}_{2}^{\prime},$$$$\frac{{u}_{ij}^{(n+1)}-{u}_{i-1,j}^{(n+1)}}{{h}_{1}}+\frac{{v}_{ij}^{(n+1)}-{v}_{i,j-1}^{(n+1)}}{{h}_{2}}=0.$$
- The Poisson equation with estimated velocity field components ${u}^{\prime}={u}^{(n+1)}$ and ${v}^{\prime}={v}^{(n+1)}$ is solved for the revised pressure field $P={P}^{(n+1)}$.$$\begin{array}{c}\frac{1}{{h}_{1}^{2}}\left({P}_{i+1,j}^{(n+1)}-2{P}_{i,j}^{(n+1)}+{P}_{i-1,j}^{(n+1)}\right)+\frac{1}{{h}_{2}^{2}}\left({P}_{i,j+1}^{(n+1)}-2{P}_{i,j}^{(n+1)}+{P}_{i,j-1}^{(n+1)}\right)=\\ =\frac{1}{\delta t}\left(\frac{{u}_{ij}^{(n+1)}-{u}_{i-1,j}^{(n+1)}}{{h}_{1}}+\frac{{v}_{ij}^{(n+1)}-{v}_{i,j-1}^{(n+1)}}{{h}_{2}}\right)+\\ +\frac{1}{{h}_{1}}\left(-{u}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i-1,j}^{(n+1)}}{{h}_{1}}\right)-{v}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i,j-1}^{(n+1)}}{{h}_{2}}\right)\right)+\\ +{\displaystyle \frac{1}{Re{h}_{1}}}\left(\frac{{u}_{i+1,j}^{(n+1)}-2{u}_{ij}^{(n+1)}+{u}_{i-1,j}^{(n+1)}}{{h}_{1}^{2}}+\frac{{u}_{i,j+1}^{(n+1)}-2{u}_{ij}^{(n+1)}+{u}_{i,j-1}^{(n+1)}}{{h}_{2}^{2}}\right)+\\ +\frac{1}{{h}_{2}}\left(-{u}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i-1,j}^{(n+1)}}{{h}_{1}}\right)-{v}_{ij}^{(n+1)}\left(\frac{{v}_{ij}^{(n+1)}-{v}_{i,j-1}^{(n+1)}}{{h}_{2}}\right)\right)+\\ +{\displaystyle \frac{1}{Re{h}_{2}}}\left(\frac{{v}_{i+1,j}^{(n+1)}-2{v}_{ij}^{(n+1)}+{v}_{i-1,j}^{(n+1)}}{{h}_{1}^{2}}+\frac{{v}_{i,j+1}^{(n+1)}-2{v}_{ij}^{(n+1)}+{v}_{i,j-1}^{(n+1)}}{{h}_{2}^{2}}\right).\end{array}$$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MGM | Multigrid Method |

SLAE | Systems of Linear Algebraic Equations |

AMG | Algebraic Multigrid |

HSS | Hermitian/Skew-Hermitian Splitting |

STS | Skew-Hermitian Triangular Splitting |

PDE | Partial Differential Equations |

CFD | Computational Fluid Dynamics |

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**Table 1.**Algebraic multigrid (AMG)+(HSS) Hermitian/skew-Hermitian splitting iterations with different $\nu $.

Grid | $\mathit{\nu}={10}^{-1}$ | $\mathit{\nu}={10}^{-2}$ | $\mathit{\nu}={10}^{-3}$ | $\mathit{\nu}={10}^{-4}$ | $\mathit{\nu}={10}^{-5}$ |
---|---|---|---|---|---|

$60\times 60$ | 25 (21.20) | 26 (21.59) | 29 (26.20) | 30 (26.54) | 21 (14.15) |

$120\times 120$ | 40 (64.50) | 45 (67.70) | 54 (94.60) | 40 (59.20) | 37 (51.50) |

$180\times 180$ | 54 (152.61) | 50 (151.52) | 64 (161.82) | 49 (126.85) | 35 (97.51) |

$260\times 260$ | 85 (192.70) | 93 (197.20) | 82 (191.52) | 83 (197.26) | 58 (126.7) |

$520\times 520$ | 90 (282.51) | 97 (290.58) | 90 (286.26) | 92 (286.85) | 85 (252.21) |

Grid | $\mathit{\nu}={10}^{-1}$ | $\mathit{\nu}={10}^{-2}$ | $\mathit{\nu}={10}^{-3}$ | $\mathit{\nu}={10}^{-4}$ | $\mathit{\nu}={10}^{-5}$ |
---|---|---|---|---|---|

$60\times 60$ | 26 (32.57) | 46 (42.51) | 54 (114.85) | n.c. | n.c. |

$120\times 120$ | 57 (83.82) | 64 (122.61) | 83 (160.50) | n.c. | n.c. |

$180\times 180$ | 59 (162.36) | 82 (185.38) | 85 (192.20) | n.c. | n.c. |

$260\times 260$ | n.c. | n.c. | n.c. | n.c. | n.c. |

$520\times 520$ | n.c. | n.c. | n.c. | n.c. | n.c. |

Grid | $\mathit{\alpha}=0.2$ | $\mathit{\alpha}=0.3$ | $\mathit{\alpha}=0.4$ | $\mathit{\alpha}=0.6$ | $\mathit{\alpha}=0.8$ | $\mathit{\alpha}=0.9$ | $\mathit{\alpha}=1.0$ |
---|---|---|---|---|---|---|---|

$60\times 60$ | 29 (26.84) | 24 (21.51) | 21 (14.15) | 42 (40.61) | 54 (58.86) | 56 (68.22) | 65 (84.20) |

$120\times 120$ | 40 (61.50) | 39 (64.67) | 37 (51.50) | 45 (67.52) | 56 (94.60) | 57 (94.20) | 82 (162.85) |

$180\times 180$ | 52 (114.2) | 35 (97.51) | 42 (129.20) | 67 (14.82) | 84 (165.84) | 86 (175.21) | 91 (196.21) |

$260\times 260$ | 58 (126.7) | 65(171.58) | 65 (187.21) | 82 (192.64) | 84 (194.54) | 91 (194.60) | 95 (197.22) |

$520\times 520$ | 82 (251.26) | 84(251.84) | 85 (252.21) | 92 (260.52) | 93 (262.42) | 94 (282.52) | 97 (290.21) |

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**MDPI and ACS Style**

Muratova, G.; Martynova, T.; Andreeva, E.; Bavin, V.; Wang, Z.-Q.
Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers. *Symmetry* **2020**, *12*, 233.
https://doi.org/10.3390/sym12020233

**AMA Style**

Muratova G, Martynova T, Andreeva E, Bavin V, Wang Z-Q.
Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers. *Symmetry*. 2020; 12(2):233.
https://doi.org/10.3390/sym12020233

**Chicago/Turabian Style**

Muratova, Galina, Tatiana Martynova, Evgeniya Andreeva, Vadim Bavin, and Zeng-Qi Wang.
2020. "Numerical Solution of the Navier–Stokes Equations Using Multigrid Methods with HSS-Based and STS-Based Smoothers" *Symmetry* 12, no. 2: 233.
https://doi.org/10.3390/sym12020233