计算物理 ›› 2014, Vol. 31 ›› Issue (4): 390-402.

• 研究论文 • 上一篇    下一篇

SAMR网格上扩散方程有限体格式的逼近性与两层网格算法

舒适1, 岳孝强1, 周志阳1, 徐小文2   

  1. 1. 湘潭大学数学与计算科学学院, 湖南 湘潭 411105;
    2. 北京应用物理与计算数学研究所, 北京 100094
  • 收稿日期:2013-08-11 修回日期:2013-12-02 出版日期:2014-07-25 发布日期:2014-07-25
  • 通讯作者: 徐小文,E-mail:xwxu@iapcm.ac.cn
  • 作者简介:舒适(1962-),男,湖南,博士,教授,博导,从事偏微分方程数值解和多重网格算法研究,E-mail:shushi@xtu.edu.cn
  • 基金资助:
    国家自然科学基金(10935003,61033009,91130002);973项目(2011CB309702);湖南省研究生科研创新项目(CX2013B255);高等学校博士学科点专项科研基金(20124301110003)资助项目

Approximation and Two-level Algorithm of Finite Volume Schemes for Diffusion Equations with Structured AMR

SHU Shi1, YUE Xiaoqiang1, ZHOU Zhiyang1, XU Xiaowen2   

  1. 1. School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China;
    2. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
  • Received:2013-08-11 Revised:2013-12-02 Online:2014-07-25 Published:2014-07-25

摘要: 针对结构自适应加密网格(SAMR)上扩散方程的求解,分析几种有限体格式的逼近性,同时设计和分析一种两层网格算法.首先,讨论一种常见的守恒型有限体格式,并给出网格加密区域和细化/粗化插值算子的条件;接着,通过在粗细界面附近引入辅助三角形单元,消除粗细界面处的非协调单元,设计了一种保对称有限体元(SFVE)格式,分析表明,该格式具有更好的逼近性,且对网格加密区域和插值算子的限制更弱;最后,为SFVE格式构造一种两层网格(TL)算法,理论分析和数值实验表明该算法的一致收敛性.

关键词: 自适应网格加密(AMR), 扩散方程, 有限体格式, 逼近性, 两层网格算法

Abstract: We analyze approximation and propose a two-level algorithm for finite volume schemes of diffusion equations with structured adaptive mesh refinement. First of all, a typically conservative finite volume scheme was discussed, along with criterion for refining and coarsening interpolation operator. Secondly, non-conforming elements around coarse-fine interface were eliminated by introducing auxiliary triangle elements. A symmetric finite volume element (SFVE) scheme was designed. And further analysis showed the scheme has better approximation. It weakens restrictions. Finally, a two-level algorithm was constructed for SFVE. Theoretical analysis and numerical experiments demonstrate uniform convergence of the algorithm.

Key words: adaptive mesh refinement (AMR), diffusion equations, finite volume schemes, approximation, two-level algorithm

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