本論文研究下述多頻高維振蕩二階微分方程系統(tǒng)的數(shù)值積分。其中M∈Rd×d是一個(gè)半正定矩陣,隱含了該系統(tǒng)的主振蕩頻率,y∈Rd,f一方面,如果（1）中右端項(xiàng)函數(shù)f依賴于y’,系統(tǒng)為一個(gè)阻尼系統(tǒng),其能量是耗散的。另一方面,如果（1）中M是一個(gè)對(duì)稱半正定矩陣且f（y）是實(shí)值函數(shù)U（y）的負(fù)梯度,那么,這就是一個(gè)哈密爾頓系統(tǒng)。此時(shí)系統(tǒng)有兩個(gè)重要性質(zhì)：辛性和能量守恒。在數(shù)值計(jì)算領(lǐng)域人們已經(jīng)認(rèn)識(shí)到,設(shè)計(jì)數(shù)值算法應(yīng)當(dāng)考慮所研究問(wèn)題的特殊結(jié)構(gòu)。近年來(lái),幾何數(shù)值積分得到了越來(lái)越多的關(guān)注。一個(gè)數(shù)值方法是幾何算法如果方法能夠精確保持系統(tǒng)的一個(gè)或多個(gè)物理／幾何結(jié)構(gòu)。更確切的說(shuō),為了使得數(shù)值算法能夠更正確反映系統(tǒng)的定性行為,要盡量保證原系統(tǒng)的基本結(jié)構(gòu)不會(huì)被數(shù)值算法破壞。因此,我們重點(diǎn)關(guān)注系統(tǒng)（1）的幾何數(shù)值積分,使得算法保持原系統(tǒng)盡可能多的結(jié)構(gòu)特性。系統(tǒng)（1）的顯著特性：振蕩性的保持將貫穿全文。本論文第一部分的研究考慮振蕩耗散系統(tǒng)。對(duì)于帶阻尼（含有y’）的振蕩二階常微分方程初值問(wèn)題,我們研究了高維ARKN方法的構(gòu)造。同時(shí),對(duì)于求解右端函數(shù)項(xiàng)依賴y和y’的一般振蕩二階常微分方程的數(shù)值方法,通過(guò)引入新的線性測(cè)試模型,... 

【文章來(lái)源】：南京大學(xué)江蘇省 211工程院校 985工程院校 教育部直屬院校

【文章頁(yè)數(shù)】：163 頁(yè)

【學(xué)位級(jí)別】：博士

【文章目錄】：
Acknowledgements
摘要
Abstract
Chapter 1 Introduction
    1.1 Numerical methods for oscillatory problems
    1.2 Conservative/dissipative systems and geometric numerical integration
    1.3 Layout of thesis
Chapter 2 Multi-frequency and multdimensionalARKN methods for general oscillatorysecond-order initial value problems
    2.1 Multi-frequency and multidimensional ARKN methods and the corre-sponding order conditions
    2.2 Novel multi-frequency and multidimensional ARKN methods for sys-tems with f depending on both y and y'
        2.2.1 Construction of multi-frequency and multidimensional ARKN methods for (2.1)
        2.2.2 Stability and phase properties of the novel multi-frequency and multidimensional ARKN methods for oscillatory system (2.1)
    2.3 Numerical experiments
    2.4 Conclusions
Chapter 3 Extended phase properties and stability analysis of RKN-type integrators for solving general oscillatory second-order initial value problems
    3.1 Motivation
    3.2 Analysis of dissipation and dispersion through a new test model
    3.3 The characteristic matrices of RKN methods and ARKN methods for general oscillatory systems
        3.3.1 RKN method
        3.3.2 ARKN methods
    3.4 Numerical experiments
    3.5 Conclusions
Chapter 4 High-order symplectic and symmetric composition methods for multi-frequency and multidimensional oscillatory Hamiltonian systems
    4.1 Motivation
    4.2 Composition of ARKN integrators for multi-frequency oscillatory sys-tem
    4.3 Composition of ERKN integrators
    4.4 Numerical experiments
    4.5 Conclusions
Chapter 5 An extended discrete gradient formula for oscillatory Hamiltonian systems
    5.1 Motivation
    5.2 Preliminaries
    5.3 An extended discrete gradient formula based on ERKN integrators and its properties
    5.4 Convergence of the the fixed-point iteration for the implicit scheme in the discrete gradient formula
    5.5 Numerical experiments
    5.6 Conclusions
Chapter 6 A linearly-fitted conservative(dissipative)scheme for conservative(dissi-pative)nonlinear wave partial differential equations
    6.1 Motivation
    6.2 Preliminaries
    6.3 Extended discrete gradient method and its remedy
    6.4 Numerical experiments
        6.4.1 Evaluation of the AVF and choice of starting approximations for fixed-point iteration
        6.4.2 Conservative wave equations
        6.4.3 Dissipative wave equations
    6.5 Conclusions
Bibliography
Foundations and publications



本文編號(hào)：3300945