【摘要】：近些年,隨著隨機(jī)力學(xué)理論的發(fā)展,隨機(jī)Hamilton系統(tǒng)受到許多學(xué)者們的關(guān)注。作為確定性Hamilton系統(tǒng)的推廣,隨機(jī)Hamilton系統(tǒng)刻畫了白噪聲影響下保守系統(tǒng)的運(yùn)動(dòng)過程。隨機(jī)Hamilton系統(tǒng)的解具有豐富的物理特性和幾何特性,比如辛性、能量守恒性、動(dòng)量守恒性等。自然地,在構(gòu)造數(shù)值方法模擬這些系統(tǒng)時(shí),一方面要求數(shù)值方法具有較高的精度和計(jì)算效率,另一方面要求數(shù)值解能夠保持系統(tǒng)的特有結(jié)構(gòu)。本文即以此為出發(fā)點(diǎn),研究了隨機(jī)分塊Runge-Kutta方法對(duì)隨機(jī)Hamilton系統(tǒng)的模擬效果,并構(gòu)造了一類兼顧辛性、能量守恒性、高收斂階的隨機(jī)分塊Runge-Kutta方法。首先,本文對(duì)一般的單一被積函數(shù)Stratonovich型分塊隨機(jī)微分方程進(jìn)行研究,應(yīng)用P級(jí)數(shù)理論分析了含單一隨機(jī)變量的隨機(jī)分塊Runge-Kutta方法的階條件。通過分析精確解和數(shù)值解的P級(jí)數(shù)展開式,以雙色根樹的形式分別給出了隨機(jī)分塊Runge-Kutta方法在均方收斂和弱收斂意義下的階條件。該部分研究表明利用單一的隨機(jī)變量可以構(gòu)造出求解單一積分函數(shù)隨機(jī)微分方程的任意階數(shù)值方法。接著,在前一部分理論研究結(jié)果基礎(chǔ)之上,研究了隨機(jī)分塊Runge-Kutta方法對(duì)保守的隨機(jī)Hamilton系統(tǒng)辛性和能量守恒性的同時(shí)保持。通過W變換構(gòu)造了一類帶有參數(shù)的隨機(jī)分塊Runge-Kutta方法(隨機(jī)參數(shù)分塊Runge-Kutta方法),證明了隨機(jī)參數(shù)分塊Runge-Kutta方法是辛的,并且在每一步求解過程中存在參數(shù)?(9)使得方法能夠保持該保守的隨機(jī)Hamilton系統(tǒng)的能量。此外,文中證明了參數(shù)?(9)仍能夠保證該方法的收斂階不變。最后選取了具有代表性的非線性保守的隨機(jī)Hamilton系統(tǒng)進(jìn)行數(shù)值試驗(yàn),驗(yàn)證了所構(gòu)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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.8

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本文編號(hào)：2259973