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抛物型方程基于POD方法的时间二阶中心差的时间二阶精度简化有限元格式

腾飞, 孙萍, 罗振东   

  1. 贵州师范大学数学与计算机科学学院, 贵阳 550001
  • 收稿日期:2010-10-30 出版日期:2011-11-15 发布日期:2011-11-15
  • 基金资助:

    国家自然科学基金(批准号: 10871022和11061009)、河北省自然科学基金(批准号: A2010001663)和贵州省科技计划项目 (批准号: 黔科合J字[2011]2367) 资助项目.

腾飞, 孙萍, 罗振东. 抛物型方程基于POD方法的时间二阶中心差的时间二阶精度简化有限元格式[J]. 计算数学, 2011, 33(4): 373-386.

Teng Fei, Sun Ping, Luo Zhendong. A REDUCED SECOND ORDER TIME ACCURATE FINITE ELEMENT FORMULATION OF TIME SECOND ORDER CENTRAL DIFFERENCE BASED ON POD APPROACH FOR PARABOLIC EQUATIONS[J]. Mathematica Numerica Sinica, 2011, 33(4): 373-386.

A REDUCED SECOND ORDER TIME ACCURATE FINITE ELEMENT FORMULATION OF TIME SECOND ORDER CENTRAL DIFFERENCE BASED ON POD APPROACH FOR PARABOLIC EQUATIONS

Teng Fei, Sun Ping, Luo Zhendong   

  1. School of Mathematics and Computer Science, Guizhou Normal University, Guiyang 550001, China
  • Received:2010-10-30 Online:2011-11-15 Published:2011-11-15
本文将特征正交分解(Proper Orthogonal Decomposition, 简记为POD)方法应用于抛物型方程通常时间二阶中心差的时间二阶精度有限元格式(简称为通常格式), 简化其为一个自由度极少但具有时间二阶精度的有限元格式, 并给出简化的时间二阶中心差的时间二阶精度有限元格式(简称为简化格式)解的误差分析. 数值例子表明在简化格式解和通常格式解之间的误差足够小的情况下, 简化格式能大大地节省自由度, 提高计算速度和计算精度,从而验证抛物型方程简化格式是可行和有效的.
A proper orthogonal decomposition (POD) method is applied to a usual second order time accurate finite element (SOTAFE) formulation of time second order central difference for parabolic equations so that it is reduced into a SOTAFE formulation of time second order central difference with fewer degrees of freedom. The errors between the reduced SOTAFE solutions of time second order central difference based on POD approach and the usual SOTAFE solutions of time second order central difference are analyzed. Numerical examples show that the reduced SOTAFE formulation of time second order central difference based POD approach can save a lot of degrees of freedom in a way that guarantees a sufficiently small errors between the reduced SOTAFE solutions of time second order central difference based on POD approach and the usual SOTAFE solutions of time second order central difference. Moreover, this verifies the reduced SOTAFE formulation of time order central difference based on POD approach is feasible and efficient solving parabolic equations.

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