几类非线性方程问题的数值解法研究
发布时间：2022-07-14 17:18
有限差分法又称网格法,是求解非线性方程的常用方法之一,该方法主要用来构造一个合理的差分格式,并且差分格式的近似解保留了原问题的一些主要性质.需要指出的是,高近似精度的差分格式并不一定能得到一个很好的近似解,因为一个合理的差分格式还必须要保留原问题固有的物理性质.因此,在保持非线性方程固有的物理律的基础上构造合理的数值求解格式是很有意义的.本文的其余部分安排如下:第一章介绍了有关研究对象的基础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方程的数值解,构造了一个保持原有守恒性质的拟紧致C-N守恒格式.该格式基于差分方法,用Brouwer不动点定理证明了解的存在性.应用能量方法证明了格式的无条件稳定性,二阶收敛性以及先验估计.数值算例验证了理论结果.第三章,基于高精度差分方法提出了一个三层的线性隐的守恒数值格式求解GRLW方程的初边值问题,对该格式包括收敛性结果作了细致地分析.数值例子表明该格式是有效的、可靠的,并且具有高精度.第四章,针对所要研究的问题,基于所研究的系统保持能量守恒性质和差分方法提出一个新的高阶有效的数值格式求解...
【文章页数】：100 页
【学位级别】：博士
【文章目录】：
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
【参考文献】：
期刊论文
[1]一维非线性Schrdinger方程的两个无条件收敛的守恒紧致差分格式[J]. 王廷春,郭柏灵. 中国科学:数学. 2011(03)
本文编号：3661549
【文章页数】：100 页
【学位级别】：博士
【文章目录】：
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
【参考文献】：
期刊论文
[1]一维非线性Schrdinger方程的两个无条件收敛的守恒紧致差分格式[J]. 王廷春,郭柏灵. 中国科学:数学. 2011(03)
本文编号：3661549
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