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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (1): 145-158.doi: 10.3969/j.issn.1005-3085.2015.01.014

• • 上一篇    

求解Burger's方程的两水平有限差分方法(英)

祖丽胡玛尔·卡迪尔1,2,   李  宁2,   黄鹏展2,   冯新龙2   

  1. 1- 喀什师范学院数学系，喀什 844006
    2- 新疆大学数学与系统科学学院，乌鲁木齐 830046
  • 收稿日期:2013-06-28 接受日期:2013-12-30 出版日期:2015-02-15 发布日期:2015-04-15
  • 基金资助:
    国家自然科学基金项目 (61163027)；教育部重点项目 (212197)；中国博士后科学基金 (2013M530438).

A Two-level Finite Difference Method for Burger's Equation

Zulhumar Kadir1,2,   LI Ning2,  HUANG Peng-zhan2,  FENG Xin-long2   

  1. 1- Department of Mathematics, Kashgar Teacher's College, Kashgar 844006
    2- College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046
  • Received:2013-06-28 Accepted:2013-12-30 Online:2015-02-15 Published:2015-04-15
  • Supported by:
    The National Natural Science Foundation of China (61163027); the Key Project of Chinese Ministry of Education (212197); the China Postdoctoral Science Foundation (2013M530438).

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摘要： 本文中提出了求解Burger's方程的两水平方法．新方法只需在粗网格上求解一个网格步长为$H$的非线性问题，在细网格上求解一个网格步长为$h$的线性问题．新格式是隐式无条件稳定的，并且能够得到与单水平解相同的收敛阶．由于单水平方法在细网格上求解一个大型非线性问题，所以我们的方法可以节省大量的计算时间．

关键词: Burger's方程, 两水平格式, 线性化逼近, Crank-Nicolson格式, 有限差分方法

Abstract:

In this paper, a two-level finite difference scheme is presented for the numerical approximation of Burger's equation. The full nonlinear problem is solved on a coarse grid of size $H$, and a linear problem is solved on a fine mesh with mesh size $h$. The new difference scheme, which is the implicit one with unconditional stability and easy computation. The method we study provides an approximate solution with nearly the same error as the usual one-level solution, which involves solving one large nonlinear problem on a fine mesh with mesh size $h$. Hence, our method is capable of significantly saving computational time.

Key words: Burger's equation, two-level scheme, linearization approximation, linearized Crank-Nicolson scheme, finite difference method

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