幾類非線性方程問題的數(shù)值解法研究
發(fā)布時(shí)間：2022-07-14 17:18
有限差分法又稱網(wǎng)格法,是求解非線性方程的常用方法之一,該方法主要用來構(gòu)造一個(gè)合理的差分格式,并且差分格式的近似解保留了原問題的一些主要性質(zhì).需要指出的是,高近似精度的差分格式并不一定能得到一個(gè)很好的近似解,因?yàn)橐粋�(gè)合理的差分格式還必須要保留原問題固有的物理性質(zhì).因此,在保持非線性方程固有的物理律的基礎(chǔ)上構(gòu)造合理的數(shù)值求解格式是很有意義的.本文的其余部分安排如下:第一章介紹了有關(guān)研究對(duì)象的基礎(chǔ)背景和本文所做的主要工作.第二章利用有限差分方法研究了Rosenau-KdV方程耦合Rosenau-RLW方程的數(shù)值解,構(gòu)造了一個(gè)保持原有守恒性質(zhì)的擬緊致C-N守恒格式.該格式基于差分方法,用Brouwer不動(dòng)點(diǎn)定理證明了解的存在性.應(yīng)用能量方法證明了格式的無條件穩(wěn)定性,二階收斂性以及先驗(yàn)估計(jì).數(shù)值算例驗(yàn)證了理論結(jié)果.第三章,基于高精度差分方法提出了一個(gè)三層的線性隱的守恒數(shù)值格式求解GRLW方程的初邊值問題,對(duì)該格式包括收斂性結(jié)果作了細(xì)致地分析.數(shù)值例子表明該格式是有效的、可靠的,并且具有高精度.第四章,針對(duì)所要研究的問題,基于所研究的系統(tǒng)保持能量守恒性質(zhì)和差分方法提出一個(gè)新的高階有效的數(shù)值格式求解...
【文章頁(yè)數(shù)】：100 頁(yè)
【學(xué)位級(jí)別】：博士
【文章目錄】：
摘要
ABSTRACT
Chapter 1 Preface
Chapter 2 Numerical analysis of a pseudo-compact C-N conser-vative scheme for the Rosenau-KdV equation coupling with the Rosenau-RLW equation
2.1 Introduction
2.2 A pseudo-compact C-N conservative scheme and its discrete con-servative invariant
2.3 Estimates and existence of difference solution
2.4 Convergence and stability of the scheme
2.5 Numerical experiments
Chapter 3 A high-accuracy conservative scheme for generalized regularized long-wave equation
3.1 Introduction
3.2 High-accuracy scheme and its discrete conservative law
3.3 Solvability and estimate
3.4 Convergence
3.5 Numerical experiments
Chapter 4 On the convergence of a high-accuracy compact conser-vative scheme for the modified regularized long-wave equation
4.1 Introduction
4.2 The high-accuracy compact conservative vector difference scheme
4.3 Discrete conservative property, estimate and solvability
4.4 Convergence and stability of the difference solution
4.5 Numerical experiments
4.6 Conclusion
Chapter 5 On the convergence of a high-accuracy conservative scheme for the Zakharov equations
5.1 Introduction
5.2 High-accuracy compact conservative scheme
5.3 Conservative properties and error estimates
5.4 Convergence
5.5 Numerical experiments
5.6 Conclusion
Chapter 6 Conclusion and future work
6.1 Conclusion
6.2 Future work
Reference
Appdenix PAPERS PUBLISHED DURING PH.D
Appdenix THANKS
【參考文獻(xiàn)】：
期刊論文
[1]一維非線性Schrdinger方程的兩個(gè)無條件收斂的守恒緊致差分格式[J]. 王廷春,郭柏靈. 中國(guó)科學(xué):數(shù)學(xué). 2011(03)
本文編號(hào)：3661549
【文章頁(yè)數(shù)】：100 頁(yè)
【學(xué)位級(jí)別】：博士
【文章目錄】：
摘要
ABSTRACT
Chapter 1 Preface
Chapter 2 Numerical analysis of a pseudo-compact C-N conser-vative scheme for the Rosenau-KdV equation coupling with the Rosenau-RLW equation
2.1 Introduction
2.2 A pseudo-compact C-N conservative scheme and its discrete con-servative invariant
2.3 Estimates and existence of difference solution
2.4 Convergence and stability of the scheme
2.5 Numerical experiments
Chapter 3 A high-accuracy conservative scheme for generalized regularized long-wave equation
3.1 Introduction
3.2 High-accuracy scheme and its discrete conservative law
3.3 Solvability and estimate
3.4 Convergence
3.5 Numerical experiments
Chapter 4 On the convergence of a high-accuracy compact conser-vative scheme for the modified regularized long-wave equation
4.1 Introduction
4.2 The high-accuracy compact conservative vector difference scheme
4.3 Discrete conservative property, estimate and solvability
4.4 Convergence and stability of the difference solution
4.5 Numerical experiments
4.6 Conclusion
Chapter 5 On the convergence of a high-accuracy conservative scheme for the Zakharov equations
5.1 Introduction
5.2 High-accuracy compact conservative scheme
5.3 Conservative properties and error estimates
5.4 Convergence
5.5 Numerical experiments
5.6 Conclusion
Chapter 6 Conclusion and future work
6.1 Conclusion
6.2 Future work
Reference
Appdenix PAPERS PUBLISHED DURING PH.D
Appdenix THANKS
【參考文獻(xiàn)】：
期刊論文
[1]一維非線性Schrdinger方程的兩個(gè)無條件收斂的守恒緊致差分格式[J]. 王廷春,郭柏靈. 中國(guó)科學(xué):數(shù)學(xué). 2011(03)
本文編號(hào)：3661549
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