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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 Godunov's theorem ： ウィキペディア英語版 Godunov's theorem In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solution of partial differential equations. The theorem states that: :''Linear numerical schemes for solving partial differential equations (PDE's), having the property of not generating new extrema (monotone scheme), can be at most first-order accurate.'' Professor Sergei K. Godunov originally proved the theorem as a Ph.D. student at Moscow State University. It is his most influential work in the area of applied and numerical mathematics and has had a major impact on science and engineering, particularly in the development of methods used in computational fluid dynamics (CFD) and other computational fields. One of his major contributions was to prove the theorem (Godunov, 1954; Godunov, 1959), that bears his name. ==The theorem== We generally follow Wesseling (2001). Aside Assume a continuum problem described by a PDE is to be computed using a numerical scheme based upon a uniform computational grid and a one-step, constant step-size, ''M'' grid point, integration algorithm, either implicit or explicit. Then if $x\_\; =\; j\backslash ,\backslash Delta\; x\; \backslash$ and $t^\; =\; n\backslash ,\backslash Delta\; t\; \backslash$, such a scheme can be described by :$$ \sum\limits_^ \varphi _^ = \sum\limits_^ . \quad \quad ( 1) In other words, the solution $\backslash varphi\; \_j^\; \backslash$ at time $n\; +\; 1$ and location $j$ is a linear function of the solution at the previous time step $n$. We assume that $\backslash beta\; \_m\; \backslash$ determines $\backslash varphi\; \_j^\; \backslash$ uniquely. Now, since the above equation represents a linear relationship between $\backslash varphi\; \_j^\; \backslash$ and $\backslash varphi\; \_j^\; \backslash$ we can perform a linear transformation to obtain the following equivalent form, :$\backslash varphi\; \_j^\; =\; \backslash sum\backslash limits\_m^\; .\; \backslash quad\; \backslash quad\; (\; 2)$ Theorem 1: ''Monotonicity preserving'' The above scheme of equation (2) is monotonicity preserving if and only if :$\backslash gamma\; \_m\; \backslash ge\; 0,\backslash quad\; \backslash forall\; m\; .\; \backslash quad\; \backslash quad\; (\; 3)$ ''Proof'' - Godunov (1959) Case 1: (sufficient condition) Assume (3) applies and that $\backslash varphi\; \_j^n\; \backslash$ is monotonically increasing with $j\; \backslash$. Then, because $\backslash varphi\; \_j^n\; \backslash le\; \backslash varphi\; \_^n\; \backslash le\; \backslash cdots\; \backslash le\; \backslash varphi\; \_^n$ it therefore follows that $\backslash varphi\; \_j^\; \backslash le\; \backslash varphi\; \_^\; \backslash le\; \backslash cdots\; \backslash le\; \backslash varphi\; \_^\; \backslash$ because :$$ \varphi _j^ - \varphi _^ = \sum\limits_m^ ^n } \right)} \ge 0 . \quad \quad ( 4) This means that monotonicity is preserved for this case. Case 2: (necessary condition) We prove the necessary condition by contradiction. Assume that $\backslash gamma\; \_p^^\; =\; \backslash sum\backslash limits\_m^M\; \backslash left(\; -\; \backslash varphi\; \_^\; \}\; \backslash right)\; =\; \backslash left\backslash c\}$ & \\ & \\ \end} \right . \quad \quad ( 6) Now choose $j=k-p\; \backslash$, to give 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「Godunov's theorem」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.