高振荡保守/耗散系统的指数型保结构算法

In the light of Feng Kang’s structure-preserving algorithms idea for Hamiltonian systems, this project focuses on the intensive study of exponential structure-preserving integrators for highly oscillatory problems. The exponential structure-preserving integrator can preserve the high oscillation of the original problem, and some useful results have been obtained from our previous effort. In this project, we try to make a long-term error analysis theory for general high-order exponential integrators by improving the modulated Fourier expansion which is suitable for the second-order Gaustchi method. We then investigate the construction of high-order exponential energy-preserving methods and their energy dissipation property for dissipative problems to meet the requirement of highly accurate numerical methods. Furthermore, the applicability of exponential integrators for spatial discretization approach will be analyzed in detail, which attempts to broaden their applications to PDEs. Finally, we will develop the semi-analytical (spatial continuous) ERKN type methods and consider their practical implementation issues, and explore the new semi-analytical approach to solving wave equations, which is entirely different from the classical semi-discrete and full discrete schemes. By the further study on structure-preserving algorithms for highly oscillatory problems and introducing the spatial continuity to structure-preserving property, this project is hopeful of obtaining some new developments in the field of structure-preserving algorithms of differential equations, and thus promote its further progress.

基于冯康先生关于哈密尔顿系统的保结构算法思想，本项目在申请人对于高振荡问题前期研究工作的基础上，拟展开对能够自然保持原问题高振荡特性的指数型保结构算法的深入研究。通过对适用于二阶Gaustchi型方法的调变Fourier展开的适当改进，尝试构建对一般高阶指数型方法有效的长时间误差分析理论；探讨高阶指数型保能量方法的构造及对耗散问题耗散性的保持，满足实际问题对高精度算法的需求；研究指数型方法在传统空间变量离散化方法中的适用性问题，扩展指数型方法的适用范围；发展波方程的基于空间连续性的半解析型ERKN方法及具体编程实现方式，探索有别于传统半离散和全离散方法的求解波方程的新思路。本项目通过对高振荡系统保结构算法的深入研究及引入空间连续性等新的保结构性质，有望取得保结构算法研究新的突破进展，从而促进该领域的进一步发展。

保结构算法思想已成为现代科学与工程计算的基本理念，在天体力学、量子力学、电磁学等领域中具有广泛应用。本项目重点研究几类微分方程保结构算法的构建及分析理论，其主要成果如下：1.对于具有高振荡性的一阶半线性微分方程，基于修正微分方程和连续级算法两种不同思想，构造两类不同的高阶指数型保能量方法，进一步完善指数型保结构算法的研究。2.对于具有高振荡解的Poisson系统，提出了构建函数拟合型保能量方法的统一框架，并给出两类具体保能量方法、澄清已有文献中的一些疑问；特别地，对于具有二次能量的线性Poisson系统给出不依赖拟合频率选取的指数型方法，证明该方法是保能量的。3.结合快速傅里叶变换技巧和分裂法思想，成功实现求解哈密尔顿波方程的半连续型ERKN方法，扩展指数型ERKN方法的应用范围。4.对于非封闭形式拉格朗日系统，提出了一种精确求解其隐式运动方程的非截断积分方法，进一步通过广义动量和速度之间的迭代求解成功设计出非封闭拉格朗日系统的辛算法和保能量方法，扩展保结构算法的研究范围。