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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–Wendroff method ： ウィキペディア英語版 Lax–Wendroff method The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite differences. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. Suppose one has an equation of the following form: : $\backslash frac+\backslash frac=0\backslash ,$ where ''x'' and ''t'' are independent variables, and the initial state, u(''x'', 0) is given. In the linear case, where '' f(u) = Au '', and ''A'' is a constant,〔LeVeque, Randy J. ''Numerical Methods for Conservation Laws", Birkhauser Verlag, 1992, p. 125.〕 : $u\_i^\; =\; u\_i^n\; -\; \backslash frac\; A\backslash left(u\_^\; -\; u\_^\; \backslash right)\; +\; \backslash frac\; A^2\backslash left(u\_^\; -2\; u\_^\; +\; u\_^\; \backslash right).$ This linear scheme can be extended to the general non-linear case in different ways. One of them is letting : $A(u)\; =\; f\text{'}(u)\; =\; \backslash frac$ The conservative form of Lax-Wendroff for a general non-linear equation is then : $u\_i^\; =\; u\_i^n\; -\; \backslash frac\; \backslash left(f(u\_^)\; -\; f(u\_^)\; \backslash right)\; +\; \backslash frac\; \backslash left(A\_\backslash left(f(u\_^)\; -\; f(u\_^)\backslash right)\; -\; A\_\backslash left(\; f(u\_^)-f(u\_^)\backslash right)\; \backslash right).$ where $A\_$ is the Jacobian matrix evaluated at $\backslash frac(U^n\_i\; +\; U^n\_)$. To avoid the Jacobian evaluation, use a two-step procedure. What follows is the Richtmyer two-step Lax–Wendroff method. The first step in the Richtmyer two-step Lax–Wendroff method calculates values for f(u(''x'', ''t'')) at half time steps, ''t''''n'' + 1/2 and half grid points, ''x''''i'' + 1/2. In the second step values at ''t''''n'' + 1 are calculated using the data for ''t''''n'' and ''t''''n'' + 1/2. First (Lax) steps: : $u\_^\; =\; \backslash frac(u\_^n\; +\; u\_^n)\; -\; \backslash frac(\; f(u\_^n)\; -\; f(u\_^n)\; ),$ : $u\_^=\; \backslash frac(u\_^n\; +\; u\_^n)\; -\; \backslash frac(\; f(u\_^n)\; -\; f(u\_^n)\; ).$ Second step: : $u\_i^\; =\; u\_i^n\; -\; \backslash frac\; \backslash left(f(u\_^)\; -\; f(u\_^)\; \backslash right).$ Another method of this same type was proposed by MacCormack. MacCormack's method uses first forward differencing and then backward differencing: First step: : $u\_^=\; u\_^n\; -\; \backslash frac(\; f(u\_^n)\; -\; f(u\_^n)\; ).$ Second step: : $u\_i^\; =\; \backslash frac(u\_^n\; +\; u\_^$ *) - \frac \left(f(u_^) - f(u_^) \right ). Alternatively, First step: : $u\_^=\; u\_^n\; -\; \backslash frac(\; f(u\_^n)\; -\; f(u\_^n)\; ).$ Second step: : $u\_i^\; =\; \backslash frac(u\_^n\; +\; u\_^$ *) - \frac \left(f(u_^) - f(u_^) \right ). ==References== * * Michael J. Thompson, ''An Introduction to Astrophysical Fluid Dynamics'', Imperial College Press, London, 2006. * 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–Wendroff 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.