随机哈密尔顿系统的高效保结构算法

Stochastic Hamiltonian systems have wide applications in many scientific fields, such as nonlinear acoustics, quantum mechanics and electromagnetism. They possess symplectic and multi-symplectic structures. Under appropriate boundary condition, they possess some physical conservation quantities, such as momentum and charge. They also satisfy some energy transforming law. Research of the efficient structure-preserving algorithms has important theoretical significance and application value. Stochastic symplectic and multi-symplectic schemes satisfy discrete stochastic symplectic and multi-symplectic structures and physical conservation quantities corresponding to original system. This project combines stochastic symplectic and multi-symplectic schemes, exponential fitted methods, compact schemes and splitting combination skills to construct efficient structure-preserving algorithms. First we reveal how to choose the step-sizes of stochastic symplectic and multi-symplectic schemes to make the methods wellposed. We also evaluate the influence of iterative methods to the conservation laws of this stochastic symplectic and multi-symplectic schemes. Then we analyze the convergence, structure-preserving and conservation propertyof the schemes using the stochastic Taylor expansion and martingale inequality. At last we use the efficient algorithms to solve stochastic Hamiltonian systems such as stochastic wave equation, stochastic Schrödinger equation and stochastic Maxwell equation. The project will provide new way to solve actual problems.

随机哈密尔顿系统在非线性声学、量子力学、电磁学等领域具有广泛的应用背景，这类系统具有随机辛结构和多辛结构，在合适的边界条件下，具有动量、电荷等守恒量，满足能量转移律，其高效保结构算法的研究具有重要的理论意义和应用价值。随机辛格式和多辛格式能够保持随机哈密尔顿系统的离散随机辛结构和多辛结构，在合适的边界条件下，能保持对应于原系统的离散的物理守恒量。本项目拟将随机辛格式和多辛格式、指数拟合方法、紧致格式和分裂组合技巧相结合，构造随机哈密尔顿系统的高效保结构算法。本项目首先从理论上揭示隐式随机辛格式和多辛格式的步长选取准则，以保证数值解的适定性，并定量估计迭代法对格式守恒性的影响；然后借助随机泰勒展式和鞅不等式等随机分析手段，给出保结构算法的收敛性、保结构和守恒性分析；最后利用保结构算法数值求解随机波动方程、随机薛定谔方程、随机麦克斯韦方程等随机哈密尔顿系统模型，为实际问题的求解提供新途径。

随机哈密尔顿系统在天体力学、流体力学、光学和电磁学等领域具有广泛的应用背景。这类哈密尔顿系统具有辛几何结构，在合适的边界条件下，具有能量、电荷等物理守恒量，其高效保结构算法的研究具有重要的理论意义和应用价值。. 根据指数拟合方法的振荡优势、离散梯度方法的守恒性、紧致算子的高精度和保辛格式的长时间数值稳定性，该项目将这些数值方法和技巧相结合，分别构造了随机振子方程、非线性薛定谔方程、Klein-Gordon 方程、随机Korteweg-de-Vries方程等哈密尔顿系统模型的高效保结构算法。一方面，从理论角度分析了几类保结构算法的收敛性、保结构性、数值稳定性和守恒性结果，另一方面，从实验角度利用这几类保结构算法求解具体的线性和非线性算例，模拟了高效算法的数值特性，验证了理l论分析的正确性。本项目的研究为随机哈密尔顿系统的分析计算提供了多种高效保结构算法，为实际背景问题的求解提供了新思路、新途径和科学依据。