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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 Multisymplectic integrator ： ウィキペディア英語版 Multisymplectic integrator In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrators are geometric integrators, meaning that they preserve the geometry of the problems; in particular, the numerical method preserves energy and momentum in some sense, similar to the partial differential equation itself. Examples of multisymplectic integrators include the Euler box scheme and the Preissman box scheme. == Multisymplectic equations == A partial differential equation (PDE) is said to be a multisymplectic equation if it can be written in the form :$Kz\_t\; +\; Lz\_x\; =\; \backslash nabla\; S(z),$ where $z(t,x)$ is the unknown, $K$ and $L$ are (constant) skew-symmetric matrices and $\backslash nabla\; S$ denotes the gradient of $S$.〔, p. 1374; , p. 335–336.〕 This is a natural generalization of $Jz\_t\; =\; \backslash nabla\; H(z)$, the form of a Hamiltonian ODE.〔, p. 186.〕 Examples of multisymplectic PDEs include the nonlinear Klein–Gordon equation $u\_\; -\; u\_\; =\; V\text{'}(u)$, or more generally the nonlinear wave equation $u\_\; =\; \backslash partial\_x\; \backslash sigma\text{'}(u\_x)\; -\; f\text{'}(u)$,〔, p. 335.〕 and the KdV equation $u\_t\; +\; uu\_x\; +\; u\_\; =\; 0$.〔, p. 339–340.〕 Define the 2-forms $\backslash omega$ and $\backslash kappa$ by :$\backslash omega(u,v)\; =\; \backslash langle\; Ku,\; v\; \backslash rangle\; \backslash quad\backslash text\backslash quad\; \backslash kappa(u,v)\; =\; \backslash langle\; Lu,\; v\; \backslash rangle$ where $\backslash langle\; \backslash ,\backslash cdot\backslash ,\; ,\; \backslash ,\backslash cdot\backslash ,\; \backslash rangle$ denotes the dot product. The differential equation preserves symplecticity in the sense that :$\backslash partial\_t\; \backslash omega\; +\; \backslash partial\_x\; \backslash kappa\; =\; 0.$〔, p. 186; , p. 336.〕 Taking the dot product of the PDE with $u\_t$ yields the local conservation law for energy: :$\backslash partial\_t\; E(u)\; +\; \backslash partial\_x\; F(u)\; =\; 0\; \backslash quad\backslash text\backslash quad\; E(u)\; =\; S(u)\; -\; \backslash tfrac12\; \backslash kappa(u\_x,u)\; ,\backslash ,\; F(u)\; =\; \backslash tfrac12\; \backslash kappa(u\_t,u).$〔, p. 187; , p. 337–338.〕 The local conservation law for momentum is derived similarly: :$\backslash partial\_t\; I(u)\; +\; \backslash partial\_x\; G(u)\; =\; 0\; \backslash quad\backslash text\backslash quad\; I(u)\; =\; \backslash tfrac12\; \backslash omega(u\_x,u)\; ,\backslash ,\; G(u)\; =\; S(u)\; -\; \backslash tfrac12\; \backslash omega(u\_t,u).$〔 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「Multisymplectic 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.