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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 Riemann invariant ： ウィキペディア英語版 Riemann invariant Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.〔 〕 ==Mathematical theory== Consider the set of conservation equations: :$l\_i\backslash left(A\_\; \backslash frac\; +a\_\backslash frac\; \backslash right)+l\_j\; b\_j=0$ where $A\_$ and $a\_$ are the elements of the matrices $\backslash mathbf$ and $\backslash mathbf$ where $l\_$ and $b\_$ are elements of vectors. It will be asked if it is possible to rewrite this equation to :$m\_j\backslash left(\backslash beta\backslash frac\; +\backslash alpha\backslash frac\; \backslash right)+l\_j\; b\_j=0$ To do this curves will be introduced in the $(x,t)$ plane defined by the vector field $(\backslash alpha,\backslash beta)$. The term in the brackets will be rewritten in terms of a total derivative where $x,t$ are parametrized as $x=X(\backslash eta),t=T(\backslash eta)$ :$\backslash frac=T\text{'}\backslash frac+X\text{'}\backslash frac$ comparing the last two equations we find :$\backslash alpha=X\text{'}(\backslash eta),\; \backslash beta=T\text{'}(\backslash eta)$ which can be now written in characteristic form :$m\_j\backslash frac+l\_jb\_j\; =\; 0$ where we must have the conditions :$l\_iA\_=m\_jT\text{'}$ :$l\_ia\_=m\_jX\text{'}$ where $m\_j$ can be eliminated to give the necessary condition :$l\_i(A\_X\text{'}-a\_T\text{'})=0$ so for a nontrival solution is the determinant :$|A\_X\text{'}-a\_T\text{'}|=0$ For Riemann invariants we are concerned with the case when the matrix $\backslash mathbf$ is an identity matrix to form :$\backslash frac\; +a\_\backslash frac=0$ notice this is homogeneous due to the vector $\backslash mathbf$ being zero. In characteristic form the system is :$l\_i\backslash frac=0$ with $\backslash frac=\backslash lambda$ Where $l$ is the left eigenvector of the matrix $\backslash mathbf$ and $\backslash lambda\; \text{'}s$ is the characteristic speeds of the eigenvalues of the matrix $\backslash mathbf$ which satisfy :$|A\; -\backslash lambda\backslash delta\_|=0$ To simplify these characteristic equations we can make the transformations such that $\backslash frac=l\_i\backslash frac$ which form :$\backslash mu\; l\_idu\_i\; =dr\_i$ An integrating factor $\backslash mu$ can be multiplied in to help integrate this. So the system now has the characteristic form :$\backslash frac=0$ on $\backslash frac=\backslash lambda\_i$ which is equivalent to the diagonal system〔 〕 :$r\_t^k\; +\backslash lambda\_kr\_x^k=0,$ $k=1,...,N.$ The solution of this system can be given by the generalized hodograph method.〔 〕〔 〕 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「Riemann invariant」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.