保持随机Hamilton系统辛结构和能量的任意阶数值方法
发布时间：2018-10-09 16:18
【摘要】：近些年,随着随机力学理论的发展,随机Hamilton系统受到许多学者们的关注。作为确定性Hamilton系统的推广,随机Hamilton系统刻画了白噪声影响下保守系统的运动过程。随机Hamilton系统的解具有丰富的物理特性和几何特性,比如辛性、能量守恒性、动量守恒性等。自然地,在构造数值方法模拟这些系统时,一方面要求数值方法具有较高的精度和计算效率,另一方面要求数值解能够保持系统的特有结构。本文即以此为出发点,研究了随机分块Runge-Kutta方法对随机Hamilton系统的模拟效果,并构造了一类兼顾辛性、能量守恒性、高收敛阶的随机分块Runge-Kutta方法。首先,本文对一般的单一被积函数Stratonovich型分块随机微分方程进行研究,应用P级数理论分析了含单一随机变量的随机分块Runge-Kutta方法的阶条件。通过分析精确解和数值解的P级数展开式,以双色根树的形式分别给出了随机分块Runge-Kutta方法在均方收敛和弱收敛意义下的阶条件。该部分研究表明利用单一的随机变量可以构造出求解单一积分函数随机微分方程的任意阶数值方法。接着,在前一部分理论研究结果基础之上,研究了随机分块Runge-Kutta方法对保守的随机Hamilton系统辛性和能量守恒性的同时保持。通过W变换构造了一类带有参数的随机分块Runge-Kutta方法(随机参数分块Runge-Kutta方法),证明了随机参数分块Runge-Kutta方法是辛的,并且在每一步求解过程中存在参数?(9)使得方法能够保持该保守的随机Hamilton系统的能量。此外,文中证明了参数?(9)仍能够保证该方法的收敛阶不变。最后选取了具有代表性的非线性保守的随机Hamilton系统进行数值试验,验证了所构造方法的保能量性。
[Abstract]:In recent years, with the development of stochastic mechanics theory, stochastic Hamilton systems have attracted many scholars' attention. As a generalization of deterministic Hamilton systems, stochastic Hamilton systems describe the motion processes of conservative systems under white noise. The solutions of stochastic Hamilton systems have abundant physical and geometric properties, such as symplectic, energy conservation, momentum conservation and so on. Naturally, in constructing numerical methods to simulate these systems, on the one hand, the numerical method is required to have higher accuracy and computational efficiency, on the other hand, the numerical solution is required to maintain the unique structure of the system. In this paper, we study the simulation effect of stochastic block Runge-Kutta method for stochastic Hamilton systems, and construct a class of stochastic block Runge-Kutta methods which take symplectic property, energy conservation and high convergence order into account. Firstly, in this paper, we study the general Stratonovich block stochastic differential equations with a single integrable function, and apply the P series theory to analyze the order conditions of the stochastic block Runge-Kutta method with a single random variable. By analyzing the P series expansions of exact solutions and numerical solutions, the order conditions of stochastic block Runge-Kutta methods in the sense of mean square convergence and weak convergence are given in the form of bicolor root trees. In this part, it is shown that an arbitrary order numerical method for solving stochastic differential equations with a single integral function can be constructed by using a single random variable. Then, based on the previous theoretical results, we study the symplectic property and energy conservation of conserved stochastic Hamilton systems by using the stochastic block Runge-Kutta method. A class of random block Runge-Kutta method with parameters (random parameter block Runge-Kutta method) is constructed by W transform. It is proved that the random parameter block Runge-Kutta method is symplectic and there are parameters in each step. (9) the method can maintain the energy of the conservative stochastic Hamilton system. In addition, the parameters are proved in this paper. (9) the convergence order of the method can still be guaranteed to remain unchanged. Finally, the representative nonlinear conservative stochastic Hamilton system is selected for numerical test, which verifies the energy conservation of the proposed method.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2259973
[Abstract]:In recent years, with the development of stochastic mechanics theory, stochastic Hamilton systems have attracted many scholars' attention. As a generalization of deterministic Hamilton systems, stochastic Hamilton systems describe the motion processes of conservative systems under white noise. The solutions of stochastic Hamilton systems have abundant physical and geometric properties, such as symplectic, energy conservation, momentum conservation and so on. Naturally, in constructing numerical methods to simulate these systems, on the one hand, the numerical method is required to have higher accuracy and computational efficiency, on the other hand, the numerical solution is required to maintain the unique structure of the system. In this paper, we study the simulation effect of stochastic block Runge-Kutta method for stochastic Hamilton systems, and construct a class of stochastic block Runge-Kutta methods which take symplectic property, energy conservation and high convergence order into account. Firstly, in this paper, we study the general Stratonovich block stochastic differential equations with a single integrable function, and apply the P series theory to analyze the order conditions of the stochastic block Runge-Kutta method with a single random variable. By analyzing the P series expansions of exact solutions and numerical solutions, the order conditions of stochastic block Runge-Kutta methods in the sense of mean square convergence and weak convergence are given in the form of bicolor root trees. In this part, it is shown that an arbitrary order numerical method for solving stochastic differential equations with a single integral function can be constructed by using a single random variable. Then, based on the previous theoretical results, we study the symplectic property and energy conservation of conserved stochastic Hamilton systems by using the stochastic block Runge-Kutta method. A class of random block Runge-Kutta method with parameters (random parameter block Runge-Kutta method) is constructed by W transform. It is proved that the random parameter block Runge-Kutta method is symplectic and there are parameters in each step. (9) the method can maintain the energy of the conservative stochastic Hamilton system. In addition, the parameters are proved in this paper. (9) the convergence order of the method can still be guaranteed to remain unchanged. Finally, the representative nonlinear conservative stochastic Hamilton system is selected for numerical test, which verifies the energy conservation of the proposed method.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8
【参考文献】
相关期刊论文 前1条
1 ;CONSTRUCTION OF HIGH ORDER SYMPLECTIC PRK METHODS[J];Journal of Computational Mathematics;1995年01期
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