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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 Lax–Friedrichs method ： ウィキペディア英語版 Lax–Friedrichs method The Lax–Friedrichs method, named after Peter Lax and Kurt O. Friedrichs, is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can be described as the FTCS (forward in time, centered in space) scheme with an artificial viscosity term of 1/2. One can view the Lax–Friedrichs method as an alternative to Godunov's scheme, where one avoids solving a Riemann problem at each cell interface, at the expense of adding artificial viscosity. ==Illustration for a Linear Problem== Consider a one-dimensional, linear hyperbolic partial differential equation for $u(x,t)$ of the form: : $u\_t\; +\; au\_x\; =\; 0\backslash ,$ on the domain : $b\; \backslash leq\; x\; \backslash leq\; c,\backslash ;\; 0\; \backslash leq\; t\; \backslash leq\; d$ with initial condition : $u(x,0)\; =\; u\_0(x)\backslash ,$ and the boundary conditions : $u(b,t)\; =\; u\_b(t)\backslash ,$ : $u(c,t)\; =\; u\_c(t).\backslash ,$ If one discretizes the domain $(b,\; c)\; \backslash times\; (0,\; d)$ to a grid with equally spaced points with a spacing of $\backslash Delta\; x$ in the $x$-direction and $\backslash Delta\; t$ in the $t$-direction, we define : $u\_i^n\; =\; u(x\_i,\; t^n)\; \backslash text\; x\_i\; =\; b\; +\; i\backslash ,\backslash Delta\; x\; ,\backslash ,\; t^n\; =\; n\backslash ,\backslash Delta\; t\; \backslash text\; i\; =\; 0,\backslash ldots,N\; ,\backslash ,\; n\; =\; 0,\backslash ldots,M,$ where : $N\; =\; \backslash frac\; ,\backslash ,\; M\; =\; \backslash frac$ are integers representing the number of grid intervals. Then the Lax–Friedrichs method for solving the above partial differential equation is given by: : $\backslash frac(u\_^n\; +\; u\_^n)\}\; +\; a\backslash frac^n\}\; =\; 0$ Or, rewriting this to solve for the unknown $u\_i^,$ : $u\_i^\; =\; \backslash frac(u\_^n\; +\; u\_^n)\; -\; a\backslash frac(u\_^n\; -\; u\_^n)\backslash ,$ Where the initial values and boundary nodes are taken from : $u\_i^0\; =\; u\_0(x\_i)$ : $u\_0^n\; =\; u\_b(t^n)$ : $u\_N^n\; =\; u\_c(t^n).$ 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「Lax–Friedrichs method」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.