非定常Navier-Stokes方程基于两重网格离散的有限元并行算法

doi:10.19596/j.cnki.1001-246x.8000

计算物理 ›› 2020, Vol. 37 ›› Issue (1): 10-18.DOI: 10.19596/j.cnki.1001-246x.8000

非定常Navier-Stokes方程基于两重网格离散的有限元并行算法

丁琪, 尚月强

西南大学数学与统计学院, 重庆 400715

收稿日期:2018-11-07 修回日期:2019-01-16 出版日期:2020-01-25 发布日期:2020-01-25
通讯作者: 尚月强(1976-),教授,从事偏微分方程数值解研究,E-mail:yqshang@swu.edu.cn
作者简介:丁琪(1995-),硕士研究生,从事偏微分方程数值解研究,E-mail:17783023729@163.com
基金资助:
国家自然科学基金（11361016）、重庆市基础与前沿探索研究计划（cstc2018jcyjAX0305）和中央高校基本科研业务费专项（XDJK2018B032）资助项目

Parallel Finite Element Algorithms Based on Two-grid Discretization for Time-dependent Navier-Stokes Equations

DING Qi, SHANG Yueqiang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received:2018-11-07 Revised:2019-01-16 Online:2020-01-25 Published:2020-01-25

摘要/Abstract

摘要： 基于两重网格离散和区域分解技巧，提出三种求解非定常Navier-Stokes方程的有限元并行算法.算法的基本思想是在每一时间迭代步，在粗网格上采用Oseen迭代法求解非线性问题，在细网格上分别并行求解Oseen、Newton、Stokes线性问题以校正粗网格解.对于空间变量采用有限元离散，时间变量采用向后Euler格式离散.数值实验验证了算法的有效性.

关键词: Navier-Stokes方程, 有限元方法, 两重网格, 并行算法

Abstract: Based on two-grid discretization and domain decomposition, three finite element parallel algorithms for unsteady Navier-Stokes equations are proposed. The key idea of the algorithms is to solve nonlinear problem firstly by Oseen iteration method on a coarse grid, and then to solve Oseen, Newton or Stokes problem in parallel on a fine grid to correct the coarse grid solution at each time step, respectively. Conforming finite element pairs are used for spatial discretization and backward Euler scheme for temporal discretization. Numerical results are shown to verify effectiveness of the algorithms.

Key words: Navier-Stokes equations, finite element method, two-grid method, parallel algorithm

中图分类号:

O241.82

丁琪, 尚月强. 非定常Navier-Stokes方程基于两重网格离散的有限元并行算法[J]. 计算物理, 2020, 37(1): 10-18.

DING Qi, SHANG Yueqiang. Parallel Finite Element Algorithms Based on Two-grid Discretization for Time-dependent Navier-Stokes Equations[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2020, 37(1): 10-18.

参考文献

[1] XU J C, ZHOU A H. Local and parallel finite element algorithms based on two-grid discretizations[J]. Mathematics of Computation, 2000, 69:881-909.
[2] XU J C, ZHOU A H. Local and parallel finite element algorithms based on two-grid discretizations for nonlinear problems[J]. Advances in Computational Mathematics, 2001, 14(4):293-327.
[3] XU J C, ZHOU A H. Local and parallel finite element algorithms for eigenvalue problems[J]. Acta Mathematicae Applicatae Sinica, 2002, 18:185-200.
[4] HE Y N, XU J C, ZHOU A H, LI J. Local and parallel finite element algorithms for the Stokes problem[J]. Numerische Mathematik, 2008, (3):415-434.
[5] HE Y N, XU J C, ZHOU A H. Local and parallel finite element algorithms for the Navier-Stokes problem[J]. Journal of Computational Mathematics, 2006, 24(3):227-238.
[6] SHANG Y Q, QIN J. Parallel finite element variational multiscale algorithms for incompressible flow at high Reynolds numbers[J]. Applied Numerical Mathematics, 2017, 117:1-21.
[7] SHANG Y Q, HE Y N, KIM D W, ZHOU X J. A new parallel finite element algorithm for the stationary Navier-Stokes equations[J]. Finite Elements in Analysis and Design, 2011, 47(11):1262-1279.
[8] MA F Y, MA Y C, WO W F. Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations[J]. Applied Mathematics and Mechanics, 2007, 28(1):27-35.
[9] LUO H, SEGAWA H, VISBAL M R. An implicit discontinuous Galerkin method for the unsteady compressible Navier-Stokes equations[J]. Computers and Fluids, 2012,53(2):133-144.
[10] LI J, HE Y N, CHEN Z X. A new stabilized finite element method for the transient Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 197:22-35.
[11] ZHANG T, YANG J. A two-level finite volume method for the unsteady Navier-Stokes equations based on two local Gauss integrations[J]. Journal of Computational and Applied Mathematics, 2014, 263(8):377-391.
[12] SUN C Y, LI Q L, YANG Z C. Least-squares finite element method for unsteady stress formulation of Navier-Stokes equations[J]. Chinese Journal of Computational Physics, 2015, 32(1):13-19.
[13] SHANG Y Q. Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations[J]. Numerical Methods for Partial Differential Equations, 2013, 29(6):2025-2046.
[14] AOUSSOU J, LIN J, LERMUSIAUX P F J. Iterated pressure-correction projection methods for the unsteady incompressible Navier-Stokes equations[J]. Journal of Computational Physics 2018, 373:940-974.
[15] SHANG Y Q, HE Y N. Parallel finite element algorithms based on fully overlapping domain decomposition for time-dependent Navier-Stokes equations[J]. Chinese Journal of Computational Physics, 2011, 28(2):181-187.
[16] YANG X C, SHANG Y Q. Two-level subgrid stabilized methods for Navier-Stokes equations at high Reynolds numbers[J]. Chinese Journal of Computational Physics, 2017, 34(6):657-665.
[17] ADAMS R. Sobolev spaces[M]. New York:Academic Press Inc, 1975.
[18] HEYWOOD J G, RANNACHER R. Finite element approximation of the nonstationary Navier-Stokes problem I:Regularity of solutions and second-order error estimates for spatial discretization[J]. SIAM Journal on Numerical Analysis, 1982, 19:275-311.
[19] GIRAULT V, RAVIART P A. Finite element methods for Navier-Stokes equations:Theory and algorithms[J]. Journal of Applied Mathematics and Mechanics, 1987, 67:579.
[20] SCHATZ A H, WAHLBIN L B. Interior maximum norm estimates for finite element methods[J]. Mathematics of Computation, 1977, 31:414-442.
[21] SCHATZ A H, WAHLBIN L B. Interior maximum-norm estimates for finite element methods,part II[J]. Mathematics of Computation, 1995, 64:907-928.
[22] HECHT F. New development in Freefem++[J]. Journal of Numerical Mathematics, 2012, 20:251-265.

非定常Navier-Stokes方程基于两重网格离散的有限元并行算法

Parallel Finite Element Algorithms Based on Two-grid Discretization for Time-dependent Navier-Stokes Equations

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

作者中心

审稿中心

期刊浏览

期刊介绍

[1]	胡晓燕, 范征锋. 惯性约束聚变内爆中基于多块结构网格的高效辐射扩散并行算法[J]. 计算物理, 2022, 39(3): 277-285.
[2]	朱家莉, 尚月强. 不可压缩流的并行两水平稳定有限元算法[J]. 计算物理, 2022, 39(3): 309-317.
[3]	蔡颖, 张存波, 刘旭, 范征锋, 刘元元, 徐小文, 张爱清. 二维球坐标系中子输运方程的一种并行SN算法[J]. 计算物理, 2022, 39(2): 143-152.
[4]	李馨东, 赵英奎, 胡宗民, 姜宗林. 基于第二粘性的Navier-Stokes方程组求解正激波结构[J]. 计算物理, 2020, 37(5): 505-513.
[5]	张庆福, 姚军, 黄朝琴, 李阳, 王月英. 裂缝性介质多尺度深度学习模型[J]. 计算物理, 2019, 36(6): 665-672.
[6]	温小静, 屈瑜, 陈城钊. 双层开口环的圆二色性数值研究[J]. 计算物理, 2019, 36(3): 357-362.
[7]	张守慧, 梁栋. 二维抛物型问题的Strang型交替分段区域分裂格式[J]. 计算物理, 2018, 35(4): 413-428.
[8]	葛志昊, 吴慧丽. 体积约束的非局部扩散问题的加罚有限元方法[J]. 计算物理, 2018, 35(2): 161-168.
[9]	杨晓成, 尚月强. 大雷诺数Navier-Stokes方程的两水平亚格子模型稳定化方法[J]. 计算物理, 2017, 34(6): 657-665.
[10]	朱晓钢, 聂玉峰, 王俊刚, 袁占斌. 分数阶对流扩散方程的特征有限元方法[J]. 计算物理, 2017, 34(4): 417-424.
[11]	祁美玲, 杨琼, 王苍龙, 田园, 杨磊. 结构材料辐照损伤的分子动力学程序GPU并行化及优化[J]. 计算物理, 2017, 34(4): 461-467.
[12]	赵国忠, 蔚喜军, 郭怀民. 二维Lagrangian坐标系下可压气动方程组的间断Petrov-Galerkin方法[J]. 计算物理, 2017, 34(3): 294-308.
[13]	叶珍宝, 朱剑, 周海京. 基于曲四面体单元剖分的CN E-H TDFEM方法[J]. 计算物理, 2016, 33(6): 652-660.
[14]	叶珍宝, 周海京. 基于高阶叠层矢量基函数的E-H时域有限元方法分析谐振腔和波导结构[J]. 计算物理, 2016, 33(3): 333-340.
[15]	叶珍宝, 周海京. 高阶间断伽辽金时域有限元方法分析三维谐振腔[J]. 计算物理, 2015, 32(4): 449-454.