本文主要研究内容
作者腾飞（2019）在《有限体积元和自然边界元基于POD降阶外推方法的几个问题研究》一文中研究指出：自然界的诸多实际物理问题都可以用某种发展型偏微分方程(组)来描述,然而除了极少数发展型偏微分方程(组)能求出其解析解外,绝大多数的发展型偏微分方程(组)是无法求出其解析解的,最有效、最经济的方法是求其数值解。其中,有限体积元法和自然边界元法是两种常用的有效的数值计算方法。对于大型的实际的工程问题,当我们采用经典的数值方法离散时,会产生数以千万的未知量。在计算过程中,由于截断误差的不断积累,以至于计算到若干步后会出现浮点外溢,不收敛的情况。本文主要利用特征投影分解(Proper Orthogonal Decomposition,简记POD)方法对双曲型偏微分方程的有限体积元格式和抛物型、Sobolev型、双曲型偏微分方程的自然边界元格式做基于POD的降阶外推数值计算理论和计算方法的研究。在确保经典的数值模型具有足够高精度的前提下,这些基于POD的降阶外推有限体积元模型和自然边界元模型,可以极大地减少未知量和计算量,从而达到节省计算机存储空间和提高计算效率以及减缓截断误差积累的目的。此外,本文还利用误差估计来指导POD基的个数的选取,这些都是对现有的基于POD技术的降阶方法的改进和创新。本文共六章,主要内容包括以下四个方面:第一部分(第二章)将POD降阶外推方法与有限体积元法结合建立双曲型方程的降阶外推有限体积元格式。首先,构造了双曲型偏微分方程的时间半离散格式以及经典有限体积元法的全离散格式,给出了经典数值解的存在唯一性、稳定性和收敛性等理论分析。然后,建立基于POD方法的双曲型偏微分方程的有限体积元降阶外推格式,讨论了基于POD的降阶外推解的存在唯一性、稳定性和收敛性等理论分析,并利用误差估计来指导POD基的个数的选取。最后,用数值例子来验证理论方法的有效性和可行性。第二部分(第三章)主要是利用POD降阶外推方法对抛物型方程建立降阶外推自然边界元格式。首先,利用Newmark方法对抛物型方程进行时间半离散,并利用自然边界元法建立全离散格式,给出了经典数值解的存在唯一性、稳定性和收敛性等理论分析。然后,建立基于POD方法的抛物型偏微分方程自然边界元降阶外推格式,讨论了基于POD的降阶外推解的存在唯一性、稳定性和收敛性等理论分析。最后,用数值例子来验证理论方法的有效性和可行性。而且进一步分析了不同瞬像个数对POD降阶外推数值模型精确度的影响,并利用误差估计来指导POD基的个数的选取。第三部分(第四章)针对Sobolev型偏微分方程建立基于POD的降阶外推自然边界元法的研究。首先,构造了 Sobolev型方程的时间半离散格式和经典自然边界元法的全离散格式,分析了经典数值解的存在唯一性、稳定性和收敛性。然后,建立基于POD方法的Sobolev型偏微分方程自然边界元降阶外推格式,讨论了基于POD的降阶外推解的存在唯一性、稳定性和收敛性等理论分析。最后,用数值例子来验证理论方法的有效性和可行性。第四部分(第五章)为双曲型方程基于POD的降阶外推自然边界元法研究。首先,建立双曲型方程时间半离散格式和经典自然边界元法的全离散格式,分析了经典数值解的存在唯一性、稳定性和收敛性。然后,建立基于POD方法的双曲型偏微分方程自然边界元降阶外推格式,讨论了基于POD的降阶外推解的存在唯一性、稳定性和收敛性等理论分析。最后用数值例子验证理论的有效性和可行性。由此表明,该种方法不仅提高了时间离散的精度,而且还极大地减少了自由度和时间方向的迭代步数,从而达到减少实际计算中截断误差的积累,提高计算精度和计算效率的目的。
Abstract
zi ran jie de zhu duo shi ji wu li wen ti dou ke yi yong mou chong fa zhan xing pian wei fen fang cheng (zu )lai miao shu ,ran er chu le ji shao shu fa zhan xing pian wei fen fang cheng (zu )neng qiu chu ji jie xi jie wai ,jue da duo shu de fa zhan xing pian wei fen fang cheng (zu )shi mo fa qiu chu ji jie xi jie de ,zui you xiao 、zui jing ji de fang fa shi qiu ji shu zhi jie 。ji zhong ,you xian ti ji yuan fa he zi ran bian jie yuan fa shi liang chong chang yong de you xiao de shu zhi ji suan fang fa 。dui yu da xing de shi ji de gong cheng wen ti ,dang wo men cai yong jing dian de shu zhi fang fa li san shi ,hui chan sheng shu yi qian mo de wei zhi liang 。zai ji suan guo cheng zhong ,you yu jie duan wu cha de bu duan ji lei ,yi zhi yu ji suan dao re gan bu hou hui chu xian fu dian wai yi ,bu shou lian de qing kuang 。ben wen zhu yao li yong te zheng tou ying fen jie (Proper Orthogonal Decomposition,jian ji POD)fang fa dui shuang qu xing pian wei fen fang cheng de you xian ti ji yuan ge shi he pao wu xing 、Sobolevxing 、shuang qu xing pian wei fen fang cheng de zi ran bian jie yuan ge shi zuo ji yu PODde jiang jie wai tui shu zhi ji suan li lun he ji suan fang fa de yan jiu 。zai que bao jing dian de shu zhi mo xing ju you zu gou gao jing du de qian di xia ,zhe xie ji yu PODde jiang jie wai tui you xian ti ji yuan mo xing he zi ran bian jie yuan mo xing ,ke yi ji da de jian shao wei zhi liang he ji suan liang ,cong er da dao jie sheng ji suan ji cun chu kong jian he di gao ji suan xiao lv yi ji jian huan jie duan wu cha ji lei de mu de 。ci wai ,ben wen hai li yong wu cha gu ji lai zhi dao PODji de ge shu de shua qu ,zhe xie dou shi dui xian you de ji yu PODji shu de jiang jie fang fa de gai jin he chuang xin 。ben wen gong liu zhang ,zhu yao nei rong bao gua yi xia si ge fang mian :di yi bu fen (di er zhang )jiang PODjiang jie wai tui fang fa yu you xian ti ji yuan fa jie ge jian li shuang qu xing fang cheng de jiang jie wai tui you xian ti ji yuan ge shi 。shou xian ,gou zao le shuang qu xing pian wei fen fang cheng de shi jian ban li san ge shi yi ji jing dian you xian ti ji yuan fa de quan li san ge shi ,gei chu le jing dian shu zhi jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi 。ran hou ,jian li ji yu PODfang fa de shuang qu xing pian wei fen fang cheng de you xian ti ji yuan jiang jie wai tui ge shi ,tao lun le ji yu PODde jiang jie wai tui jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi ,bing li yong wu cha gu ji lai zhi dao PODji de ge shu de shua qu 。zui hou ,yong shu zhi li zi lai yan zheng li lun fang fa de you xiao xing he ke hang xing 。di er bu fen (di san zhang )zhu yao shi li yong PODjiang jie wai tui fang fa dui pao wu xing fang cheng jian li jiang jie wai tui zi ran bian jie yuan ge shi 。shou xian ,li yong Newmarkfang fa dui pao wu xing fang cheng jin hang shi jian ban li san ,bing li yong zi ran bian jie yuan fa jian li quan li san ge shi ,gei chu le jing dian shu zhi jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi 。ran hou ,jian li ji yu PODfang fa de pao wu xing pian wei fen fang cheng zi ran bian jie yuan jiang jie wai tui ge shi ,tao lun le ji yu PODde jiang jie wai tui jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi 。zui hou ,yong shu zhi li zi lai yan zheng li lun fang fa de you xiao xing he ke hang xing 。er ju jin yi bu fen xi le bu tong shun xiang ge shu dui PODjiang jie wai tui shu zhi mo xing jing que du de ying xiang ,bing li yong wu cha gu ji lai zhi dao PODji de ge shu de shua qu 。di san bu fen (di si zhang )zhen dui Sobolevxing pian wei fen fang cheng jian li ji yu PODde jiang jie wai tui zi ran bian jie yuan fa de yan jiu 。shou xian ,gou zao le Sobolevxing fang cheng de shi jian ban li san ge shi he jing dian zi ran bian jie yuan fa de quan li san ge shi ,fen xi le jing dian shu zhi jie de cun zai wei yi xing 、wen ding xing he shou lian xing 。ran hou ,jian li ji yu PODfang fa de Sobolevxing pian wei fen fang cheng zi ran bian jie yuan jiang jie wai tui ge shi ,tao lun le ji yu PODde jiang jie wai tui jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi 。zui hou ,yong shu zhi li zi lai yan zheng li lun fang fa de you xiao xing he ke hang xing 。di si bu fen (di wu zhang )wei shuang qu xing fang cheng ji yu PODde jiang jie wai tui zi ran bian jie yuan fa yan jiu 。shou xian ,jian li shuang qu xing fang cheng shi jian ban li san ge shi he jing dian zi ran bian jie yuan fa de quan li san ge shi ,fen xi le jing dian shu zhi jie de cun zai wei yi xing 、wen ding xing he shou lian xing 。ran hou ,jian li ji yu PODfang fa de shuang qu xing pian wei fen fang cheng zi ran bian jie yuan jiang jie wai tui ge shi ,tao lun le ji yu PODde jiang jie wai tui jie de cun zai wei yi xing 、wen ding xing he shou lian xing deng li lun fen xi 。zui hou yong shu zhi li zi yan zheng li lun de you xiao xing he ke hang xing 。you ci biao ming ,gai chong fang fa bu jin di gao le shi jian li san de jing du ,er ju hai ji da de jian shao le zi you du he shi jian fang xiang de die dai bu shu ,cong er da dao jian shao shi ji ji suan zhong jie duan wu cha de ji lei ,di gao ji suan jing du he ji suan xiao lv de mu de 。
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论文详细介绍
论文作者分别是来自华北电力大学(北京)的腾飞，发表于刊物华北电力大学(北京)2019-10-28论文，是一篇关于特征投影分解方法论文,双曲型偏微分方程论文,抛物型偏微分方程论文,型偏微分方程论文,有限体积元法论文,自然边界元法论文,华北电力大学(北京)2019-10-28论文的文章。本文可供学术参考使用，各位学者可以免费参考阅读下载，文章观点不代表本站观点，资料来自华北电力大学(北京)2019-10-28论文网站，若本站收录的文献无意侵犯了您的著作版权，请联系我们删除。
标签：特征投影分解方法论文; 双曲型偏微分方程论文; 抛物型偏微分方程论文; 型偏微分方程论文; 有限体积元法论文; 自然边界元法论文; 华北电力大学(北京)2019-10-28论文;