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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 Beeman's algorithm ： ウィキペディア英語版 Beeman's algorithm Beeman's algorithm is a method for numerically integrating ordinary differential equations of order 2, more specifically Newton's equations of motion $\backslash ddot\; x=A(x)$. It was designed to allow high numbers of particles in simulations of molecular dynamics. There is a direct or explicit and an implicit variant of the method. The direct variant was published by Schofield 〔 in 1973 as a personal communication from Beeman. This is what is commonly known as Beeman's method. It is a variant of the Verlet integration method. It produces identical positions, but uses a different formula for the velocities. Beeman〔 in 1976 published a class of implicit (predictor-corrector) multi-step methods, where Beeman's method is the direct variant of the third order method in this class. == Equation == The formula used to compute the positions at time $t\; +\; \backslash Delta\; t$ in the full predictor-corrector〔 scheme is: * Predict $x(t+\backslash Delta\; t)$ from data at times $t\backslash textt\; -\; \backslash Delta\; t$ ::$$ x(t+\Delta t) = x(t) + v(t) \Delta t + \frac\Bigl( 4 a(t) - a(t - \Delta t)\Bigr)\Delta t^2 + O( \Delta t^4) . * Correct position and velocities at time $t\; +\; \backslash Delta\; t$ from data at times $t\backslash textt+\backslash Delta\; t$ by repeated evaluation of the differential equation to get the acceleration $a(t+\backslash Delta\; t)$ and of the equations of the implicit system ::$\backslash begin$ x(t+\Delta t) &= x(t) + v(t) \Delta t + \frac\Bigl(a(t+\Delta t) + 2a(t)\Bigr)\Delta t^2 + O(\Delta t^4);\\ v(t+\Delta t)\Delta t &=x(t+\Delta t)-x(t) + \frac16 \Bigl(2a(t+\Delta t) + a(t)\Bigr)\Delta t^2 + O(\Delta t^4); \end :In tests it was found that this corrector step needs to be repeated at most twice. The values on the right are the old values of the last iterations, resulting in the new values on the left. Using only the predictor formula and the corrector for the velocities one obtains a direct or explicit method〔 which is a variant of the Verlet integration method:〔 :$\backslash begin$ x(t+\Delta t) &= x(t) + v(t) \Delta t + \frac\Bigl( 4 a(t) - a(t - \Delta t)\Bigr)\Delta t^2 + O( \Delta t^4) \\ v(t+\Delta t) &=v(t) + \frac16 \Bigl(2a(t+\Delta t) + 5a(t)-a(t-\Delta t)\Bigr)\Delta t + O(\Delta t^3); \end This is the variant that is usually understood as ''Beeman's method''. Beeman〔 also proposed to alternatively replace the velocity update in the last equation by the second order Adams–Moulton method: :$$ v(t + \Delta t) = v(t) + \frac\Bigl(5a(t + \Delta t) + 8a(t) - a(t - \Delta t)\Bigr)\Delta t + O(\Delta t^3) where *$t$ is present time (i.e.: independent variable) *$\backslash Delta\; t$ is the time step size *$x(t)$ is the position at time t *$v(t)$ is the velocity at time t *$a(t)$ is the acceleration at time t, computed as a function of $x(t)$ *the last term is the error term, using the big O notation 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「Beeman's algorithm」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.