發(fā)布時(shí)間：2018-03-04 14:47

  本文選題：高階問(wèn)題　切入點(diǎn)：新的譜和譜元方法　出處：《江蘇師范大學(xué)》2017年碩士論文　論文類(lèi)型：學(xué)位論文

【摘要】：有限差分法、有限元方法和譜方法是求解偏微分方程的三種主要方法.譜方法是以整體光滑的正交多項(xiàng)式作為基底逼近問(wèn)題的解,其優(yōu)點(diǎn)是高精度,也就是說(shuō),只要問(wèn)題的解越光滑,逼近的程度越好.在本文中,我們針對(duì)高階問(wèn)題提出一種新的譜和譜元方法.首先,對(duì)于帶齊次邊界條件的常系數(shù)的四階問(wèn)題,我們利用矩陣正交分解的技巧構(gòu)造了一類(lèi)新的基函數(shù).其次,基于新的基函數(shù),我們提出一類(lèi)新的時(shí)空譜方法,所提算法在保證高精度的同時(shí)節(jié)省了計(jì)算時(shí)間.我們證明了該方法在空間方向和時(shí)間方向上同時(shí)具有高精度,數(shù)值結(jié)果也驗(yàn)證了理論分析.最后,我們利用新的基函數(shù)的優(yōu)點(diǎn),對(duì)于一維和二維四階問(wèn)題提出了新的譜元方法的求解格式,數(shù)值結(jié)果揭示了譜元方法的有效性.
[Abstract]:The finite difference method, finite element method and spectral method are three main methods for solving partial differential equations. As long as the solution of the problem is smoother, the degree of approximation is better. In this paper, we propose a new spectral and spectral element method for higher order problems. We construct a new class of basis functions by using the technique of matrix orthogonal decomposition. Secondly, based on the new basis functions, we propose a new kind of space-time spectral method. It is proved that the proposed method has high accuracy in both spatial and temporal directions, and the numerical results verify the theoretical analysis. Finally, we use the advantages of the new basis function. A new spectral element method is proposed for solving one-dimensional and two-dimensional fourth-order problems. The numerical results show the validity of the spectral element method.
【學(xué)位授予單位】：江蘇師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O241.82

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1 姚漢峰;基于一類(lèi)非標(biāo)準(zhǔn)基底的結(jié)構(gòu)化的譜元方法及其應(yīng)用[D];江蘇師范大學(xué);2017年

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本文編號(hào)：1566088