Hamilton方程保能量數值方法的研究
發(fā)布時間：2018-08-11 09:58
【摘要】：Hamilton系統(tǒng)在天體力學、量子力學等領域有著廣泛的應用.一切真實的、耗散效應可忽略不計的物理過程都可以表達為這樣或那樣的Hamilton形式.但是,對于許多Hamilton方程,求其精確解常常是非常困難的.因此系統(tǒng)地研究Hamilton方程的數值解法具有重要的理論和實際意義.我們希望數值解法盡可能保持Hamilton系統(tǒng)的重要性質.離散梯度在構造Hamilton方程保能量數值方法中起著非常重要的作用.可利用離散梯度構造保能量數值解法,離散梯度包括坐標增量離散梯度、平均離散梯度等.本文介紹了 Hamilton方程的一些性質及辛幾何算法的相關知識,主要研究了Hamilton方程保能量數值方法.我們研究了 Hamilton函數fg,fgh的一些離散梯度之間的聯(lián)系,給出了它們的關系式.在此基礎上,給出了由一階保能量的數值解法構造二階保能量的數值解法的方法.最后通過具體的例子進行了數值模擬,數值結果顯示新的數值解法是保能量數值方法.
[Abstract]:Hamilton system is widely used in celestial mechanics, quantum mechanics and other fields. All real, dissipative physical processes can be expressed in one form or another in Hamilton form. However, for many Hamilton equations, it is often very difficult to find its exact solution. Therefore, it is of great theoretical and practical significance to study the numerical solution of Hamilton equation systematically. We hope that the numerical solution can preserve the important properties of the Hamilton system as much as possible. Discrete gradient plays an important role in constructing the energy preserving numerical method of Hamilton equation. The energy preserving numerical solution can be constructed by using discrete gradient. The discrete gradient includes coordinate increment discrete gradient, average discrete gradient and so on. In this paper, some properties of Hamilton equation and the knowledge of symplectic geometric algorithm are introduced, and the energy preserving numerical method of Hamilton equation is mainly studied. In this paper, we study the relations between some discrete gradients of Hamilton function fgfgh, and give their relations. On this basis, the method of constructing the second order energy preserving numerical solution from the first order energy preserving numerical solution is given. Finally, numerical simulation is carried out through a concrete example. The numerical results show that the new numerical method is energy preserving numerical method.
【學位授予單位】：北京交通大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8
本文編號：2176657
[Abstract]:Hamilton system is widely used in celestial mechanics, quantum mechanics and other fields. All real, dissipative physical processes can be expressed in one form or another in Hamilton form. However, for many Hamilton equations, it is often very difficult to find its exact solution. Therefore, it is of great theoretical and practical significance to study the numerical solution of Hamilton equation systematically. We hope that the numerical solution can preserve the important properties of the Hamilton system as much as possible. Discrete gradient plays an important role in constructing the energy preserving numerical method of Hamilton equation. The energy preserving numerical solution can be constructed by using discrete gradient. The discrete gradient includes coordinate increment discrete gradient, average discrete gradient and so on. In this paper, some properties of Hamilton equation and the knowledge of symplectic geometric algorithm are introduced, and the energy preserving numerical method of Hamilton equation is mainly studied. In this paper, we study the relations between some discrete gradients of Hamilton function fgfgh, and give their relations. On this basis, the method of constructing the second order energy preserving numerical solution from the first order energy preserving numerical solution is given. Finally, numerical simulation is carried out through a concrete example. The numerical results show that the new numerical method is energy preserving numerical method.
【學位授予單位】：北京交通大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8
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