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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 scheme ： ウィキペディア英語版 Godunov's scheme In numerical analysis and computational fluid dynamics, Godunov's scheme is a conservative numerical scheme, suggested by S. K. Godunov in 1959, for solving partial differential equations. One can think of this method as a conservative finite-volume method which solves exact, or approximate Riemann problems at each inter-cell boundary. In its basic form, Godunov's method is first order accurate in both space, and time, yet can be used as a base scheme for developing higher-order methods. ==Basic scheme== Following the classical Finite-volume method framework, we seek to track a finite set of discrete unknowns, : $Q^\_i\; =\; \backslash frac\; \backslash int\_\; \}\; q(t^n,\; x)\backslash ,\; dx$ where the $x\_i\; =\; x\_,"\; TITLE="x\_,">x\_\; ),$ we obtain a Method of lines (MOL) formulation for the spatial cell averages: : $\backslash frac\; Q\_i(\; t\; )\; =\; -\backslash frac\; \backslash left(\; f(\; q(\; t,\; x\_\; )\; )\; -\; f(\; q(\; t,\; x\_\; )\; )\; \backslash right),$ which is a classical description of the first order, upwinded finite volume method. (c.f. Leveque - Finite Volume Methods for Hyperbolic Problems ) Exact time integration of the above formula from time $t\; =\; t^n$ to time $t\; =\; t^$ yields the exact update formula: : $Q^\_i\; =\; Q^n\_i\; -\; \backslash frac\; \backslash int\_^\; \backslash left(\; f(\; q(\; t,\; x\_\; )\; )\; -\; f(\; q(\; t,\; x\_\; )\; )\; \backslash right)\backslash ,\; dt.$ Godunov's method replaces the time integral of each : $\backslash int\_^\; f(\; q(\; t,\; x\_\; )\; )\backslash ,\; dt$ with a Forward Euler method which yields a fully discrete update formula for each of the unknowns $Q^n\_i$. That is, we approximate the integrals with : $\backslash int\_^\; f(\; q(\; t,\; x\_\; )\; )\backslash ,\; dt\; \backslash approx\; \backslash Delta\; t\; f^\backslash downarrow\backslash left(\; Q^n\_,\; Q^n\_i\; \backslash right),$ where $f^\backslash downarrow\backslash left(\; q\_l,\; q\_r\; \backslash right)$ is an approximation to the exact solution of the Riemann problem. For consistency, one assumes that : $f^\backslash downarrow(\; q\_l\; ,\; q\_r\; )\; =\; f(\; q\_l\; )\; \backslash quad\; \backslash text\; \backslash quad\; q\_l\; =\; q\_r,$ and that $f^\backslash downarrow$ is increasing in the first argument, and decreasing in the second argument. For scalar problems where $f\text{'}(\; q\; )\; >\; 0$, one can use the simple Upwind scheme, which defines $f^\backslash downarrow(\; q\_l,\; q\_r\; )\; =\; f(\; q\_l\; )$. The full Godunov scheme requires the definition of an approximate, or an exact Riemann solver, but in its most basic form, is given by: : $Q^\_i\; =\; Q^n\_i\; -\; \backslash lambda\; \backslash left(\; \backslash hat^n\_\; -\; \backslash hat^n\_\; \backslash right),\; \backslash quad\; \backslash lambda\; =\; \backslash frac,\; \backslash quad\; \backslash hat^n\_\; =\; f^\backslash downarrow\backslash left(\; Q^n\_,\; Q^n\_i\; \backslash right)$ 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 scheme」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.