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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Geometric integrator ： ウィキペディア英語版 Geometric integrator In the mathematical field of numerical ordinary differential equations, a geometric integrator is a numerical method that preserves geometric properties of the exact flow of a differential equation. ==Pendulum example== We can motivate the study of geometric integrators by considering the motion of a pendulum. Assume that we have a pendulum whose bob has mass $m=1$ and whose rod is massless of length $\backslash ell=1$. Take the acceleration due to gravity to be $g=1$. Denote by $q(t)$ the angular displacement of the rod from the vertical, and by $p(t)$ the pendulum's momentum. The Hamiltonian of the system, the sum of its kinetic and potential energies, is :$H(q,p)\; =\; T(p)+U(q)\; =\; \backslash fracp^2\; -\; \backslash cos\; q,$ which gives Hamilton's equations :$(\backslash dot\; q,\backslash dot\; p)\; =\; (p,-\backslash sin\; q).\; \backslash ,$ It is natural to take the configuration space $Q$ of all $q$ to be the unit circle $\backslash mathbb\; S^1$, so that $(q,p)$ lies on the cylinder $\backslash mathbb\; S^1\backslash times\backslash mathbb\; R$. However, we will take $(q,p)\backslash in\backslash mathbb\; R^2$, simply because $(q,p)$-space is then easier to plot. Define $z(t)\; =\; (q(t),p(t))^$ and $f(z)\; =\; (p,-\backslash sin\; q)^$. Let us experiment by using some simple numerical methods to integrate this system. As usual, we select a constant step size, $h$, and for an aribtrary non-negative integer $k$ we write $z\_k:=z(kh)$. We use the following methods. : $z\_\; =\; z\_k\; +\; hf(z\_k)\; \backslash ,$ (explicit Euler), : $z\_\; =\; z\_k\; +\; hf(z\_)\; \backslash ,$ (implicit Euler), : $z\_\; =\; z\_k\; +\; hf(q\_k,p\_)\; \backslash ,$ (symplectic Euler), : $z\_\; =\; z\_k\; +\; hf((z\_+z\_k)/2)\; \backslash ,$ (implicit midpoint rule). (Note that the symplectic Euler method treats ''q'' by the explicit and $p$ by the implicit Euler method.) The observation that $H$ is constant along the solution curves of the Hamilton's equations allows us to describe the exact trajectories of the system: they are the level curves of $p^2/2\; -$ \cos q. We plot, in $\backslash mathbb\; R^2$, the exact trajectories and the numerical solutions of the system. For the explicit and implicit Euler methods we take $h=0.2$, and ''z''0 = (0.5, 0) and (1.5, 0) respectively; for the other two methods we take $h=0.3$, and ''z''0 = (0, 0.7), (0, 1.4) and (0, 2.1). The explicit (resp. implicit) Euler method spirals out from (resp. in to) the origin. The other two methods show the correct qualitative behaviour, with the implicit midpoint rule agreeing with the exact solution to a greater degree than the symplectic Euler method. Recall that the exact flow $\backslash phi\_t$ of a Hamiltonian system with one degree of freedom is area-preserving, in the sense that :$\backslash det\backslash frac\; =\; 1$ for all $t$. This formula is easily verified by hand. For our pendulum example we see that the numerical flow $\backslash Phi\_,h\}:z\_k\backslash mapsto\; z\_$ of the explicit Euler method is not area-preserving; viz., :$\backslash det\backslash frac\backslash Phi\_,h\}(z\_0)$ = \begin1&h\\-h\cos q_0&1\end = 1+h^2\cos q_0. A similar calculation can be carried out for the implicit Euler method, where the determinant is :$\backslash det\backslash frac\backslash Phi\_,h\}(z\_0)$ = (1+h^2\cos q_1)^. However, the symplectic Euler method is area-preserving: :$$ \begin1&-h\\0&1\end\frac\Phi_,h}(z_0) = \begin1&0\\-h\cos q_0&1\end, thus $\backslash det(\backslash partial\backslash Phi\_,h\}/\backslash partial\; (q\_0,p\_0))\; =\; 1$. The implicit midpoint rule has similar geometric properties. To summarize: the pendulum example shows that, besides the explicit and implicit Euler methods not being good choices of method to solve the problem, the symplectic Euler method and implicit midpoint rule agree well with the exact flow of the system, with the midpoint rule agreeing more closely. Furthermore, these latter two methods are area-preserving, just as the exact flow is; they are two examples of geometric (in fact, symplectic) integrators. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric integrator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.