振荡微分方程的几何数值积分理论与算法

The numerical solution to oscillatory differential equations has been an active research field in recent years, which has extensive applications in fields of modern science such as celestial mechanics and quantum mechanics. This project focuses on theory and numerical algorithms which can preserve the oscillatory character of the exact solution. Theoretical work will be carried out in three successively deepening aspects: 1) Theories of rooted trees, B-series and algebraic order conditions for Runge-Kutta(-Nystrom)（RK(N)） methods based on the propagation of the inner-stage errors; 2）Stability theory of RKN methods for stiff oscillatory systems; 3) Phase properties and stability theory for multi-frequency fitted multi-step methods. Four types of practical algorithms will be constructed in accordance with specific structures of the oscillatory differential equations: 1) Symplectic and symmetric extended RKN (ERKN) methods with step controlling for the high-dimensional oscillatory systems obtained by semi-discreatizing partial differential equations; 2）Energy-preserving ERKN methods for general high-dimensional oscillatory Hamiltonian systems; 3) Diagonally implicit ERKN methods for stiffly oscillatory problems; 4) Multi-frequency trigonometrically fitted P-stable multi-step methods for problems with several principal frequencies. The high effectiveness of the newly developed algorithms will be tested by applying them to oscillatory problems in sciences and engineering such as the quantum Schrodinger equation.

振荡微分方程数值解在天体力学、量子力学等现代科学许多领域中有着广泛的应用。本项目研究能够保持精确解振荡性质的数值算法及其理论。理论研究包括逐步深入的三部分：1）基于内级误差传播的Runge-Kutta(-Nystrom)(RK(N))型方法的根树理论、B-级数理论、代数阶条件理论；2）刚性振荡系统的RKN方法的稳定性理论；3）基于多频拟合的多步方法的相性质和稳定性理论。项目将针对不同结构的振荡微分方程构建四类实用算法：1）偏微分方程半离散化导致的高维振荡系统的变步长的辛与对称扩展的RKN（ERKN）方法；2）一般高维振荡哈密尔顿系统的保能量ERKN方法；3）刚性振荡问题的对角隐式ERKN方法；4）具有多个主频率的振荡问题的P稳定多频三角拟合多步方法及振荡频率的估计问题。项目将通过科学和工程中常见的量子薛定谔方程等振荡问题的实际计算，测试新算法的高效性。

本项目研究振荡微分方程的保结构数值求解理论与算法。主要研究成果如下：为了求解薛定谔方程，构造了具有优化的相误差的数值方法，包括高阶两导数Runge-Kutta方法，指数拟合多导数Obrechkoff单步方法，三角拟合两步混合方法；对形如Ay''+By=f(y)非线性振荡Hamilton系统, 构建了保能量的GAAVF方法；针对高振荡微分方程给出了三角拟合两步混合方法，考虑了内级误差对更新级的影响；针对一阶振荡微分方程，构造了一类新的指数Fourier配置方法。借助于常数变异公式，延续了局部Fourier展开方法, 分析了这类方法保持二次不变量和Hamilton量的精度。结合阶条件，通过选取适当的参数可以使得方法的相误差阶达到最大；FSAL性质和变步长嵌入技术有利于提高算法的效率。大量的数值实验结果表明，本项目建立的新算法在计算精度、计算效率、保持振荡系统相结构以及保持Hamilton量方面在不同程度地超过了文献中的方法。这些新算法的开发对量子力学、物理学、化学、生物学和工程中振荡微分方程的高效数值计算具有重要的应用价值。