哈密尔顿系统的高效的辛和多辛算法

From the standpoint of simplicity,high efficiency and practicality of a numerical algorithm, we study the symplectic and multisymplectic algorithms for deterministic Hamiltonian systems and stochastic Hamiltonian systems in the work.Firstly, we will discuss the symplectic and multisymplectic algorithms for determinstic Hamiltonian systems: splitting symplectic and multisymplectic algorithms,extrapolation symplectic and multisymplectic algorithms. Moreover, we will analyze the methods numerically. Based on the theoretical analysis, we will study their application to quantum mechanics, electromagnetism,etc. And then, we will study pseudo-symplectic and pseudo-multisymplectic algorithms to improve the computational efficiency of numerical methods with the characters of symplectic or multisymplectic algorithms. In addition, their numerical analysis and application will be investigated. Finally, the stochastic symplectic and multisymplectic algorithms will be studied for stochastic Hamiltonian systems, espcially the latter. It consists of stochastic symplectic algorithms, the multisymplectic structure and theory for stochastic Hamiltonian systems, and how to construct stochastic multisymplectic algorithms. It will discuss the degree preserved of some conserved quanties by the methods, strong (weak) convergence and stability of the methods. Furthermore, their application to specific stochastic problems will be considered, such as stochastic Maxwell equations,stochastic Schr?dinger equations, stochastic KdV equations.

本项目从数值算法应该简单、高效、实用的角度出发，研究确定性哈密尔顿系统和随机哈密尔顿系统的辛和多辛算法。首先，讨论确定性哈密尔顿系统的辛和多辛算法，主要有分裂步辛和多辛算法、外推辛和多辛算法。同时对算法进行数值分析，进而研究这些算法和理论在量子力学、电磁理论等领域中的应用。其次，研究哈密尔顿系统的伪辛和伪多辛算法，以提高具有辛或多辛算法特性的数值算法的计算效率。对算法进行数值分析并应用于数值计算。最后，研究随机哈密尔顿系统的随机辛和多辛算法，特别是随机多辛算法。主要包括随机辛算法、随机哈密尔顿系统的多辛结构和理论，随机多辛算法的构造方法，讨论算法对一些守恒量的保持情况、强(弱)收敛性和稳定性等。同时研究其在具体随机辛和多辛哈密尔顿系统中的应用，如随机Maxwell方程、随机Schr?dinger方程、随机KdV方程。

项目主要研究了动力系统的高效的保结构算法，主要有哈密尔顿动力系统的保持辛和多辛几何结构的保结构算法，保能量等不变量的保结构算法，保随机哈密尔顿系统的保结构算法。项目研究的出发点是数值方法不仅要稳定、收敛、相容、长时间数模拟能力，还应该高效、实用、易于执行等。首先，我们研究了确定性哈密尔顿系统高效、实用、易于编写代码的具有辛或多辛算法数值特征的保结构算法。把这些算法应用到了量子力学、超导、凝聚态物理等领域的数学物理模型的数值模拟研究，讨论了一些有趣的物理现象。其次，研究了哈密尔顿系统的外推多辛算法的构造方法和应用。把外推技术与多辛算法结合起来，构造了高精度、实用、便于编写程序的高精度多辛算法。再次，研究了动力系统高效、高精度的保能量结构的数值格式，对格式进行了理论分析和数值模拟实验。最后，研究了随机哈密尔顿系统高效、高精度的保结构算法，特别是保能量、保质量的数值格式，并数值模拟了各类物理现象，考察了了模型参数对解的影响作用。