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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 problem ： ウィキペディア英語版 Riemann problem A Riemann problem, named after Bernhard Riemann, consists of an initial value problem composed of a conservation equation together with piecewise constant data having a single discontinuity. The Riemann problem is very useful for the understanding of equations like Euler conservation equations because all properties, such as shocks and rarefaction waves, appear as characteristics in the solution. It also gives an exact solution to some complex nonlinear equations, such as the Euler equations. In numerical analysis, Riemann problems appear in a natural way in finite volume methods for the solution of conservation law equations due to the discreteness of the grid. For that it is widely used in computational fluid dynamics and in MHD simulations. In these fields Riemann problems are calculated using Riemann solvers. ==The Riemann problem in linearized gas dynamics== As a simple example, we investigate the properties of the one dimensional Riemann problem in gas dynamics, which initial condition is defined by : $$ \begin \rho \\ u \end = \begin \rho_L \\ u_L\end \text x \leq 0 \qquad \text \qquad \begin \rho \\ u \end = \begin \rho_R \\ u_R \end \text x > 0 where ''x'' = 0 separates two different states, together with the linearised gas dynamic equation (see gas dynamics for derivation). : $$ \begin \frac + \rho_0 \frac & = 0 \\() \frac + \frac \frac & = 0 \end where we can assume w.l.o.g. $a\backslash ge\; 0$. We now can rewrite the above equation in conservative form $U\_t\; +\; A\; \backslash cdot\; U\_x\; =\; 0$: : $$ U = \begin \rho \\ u \end, \quad A = \begin 0 & \rho_0 \\ \frac & 0 \end The eigenvalues of the system are the characteristics of the system $\backslash lambda\_1\; =\; -a,\; \backslash lambda\_2\; =\; a$. They give the propagation speed of the medium, including that of any discontinuity, which is the speed of sound here. The corresponding eigenvectors are : $$ \mathbf^ = \begin \rho_0 \\ -a \end, \quad \mathbf^ = \begin \rho_0 \\ a \end. By decomposing the left state $u\_L$ in terms of the eigenvectors, we get for some $\backslash alpha\_,\backslash alpha\_$ : $$ U_L = \begin \rho_L \\ u_L \end = \alpha_1\mathbf^ + \alpha_2 \mathbf^ . Now we can solve for $\backslash alpha\_1$ and $\backslash alpha\_2$: : $$ \begin \alpha_1 & = \frac \\() \alpha_2 & = \frac \end Analogously :$U\_R\; =\; \backslash begin\; \backslash rho\_R\; \backslash \backslash \; u\_R\; \backslash end\; =\; \backslash beta\_1\backslash mathbf^+\backslash beta\_2\backslash mathbf^$ for : $$ \begin \beta_1 & = \frac \\() \beta_2 & = \frac \end Using this, in the domain in between the two characteristics $t=a|x|$, we get the final constant solution : $$ U_ * = \begin \rho_ * \\ u_ * \end =\beta_1\mathbf^+\alpha_2\mathbf^ = \beta_1 \begin \rho_0 \\ -a\end + \alpha_2 \begin \rho_0 \\ a \end and the (piecewise constant) solution in the entire domain $t>0$: :$U(t,x)$ = \begin \rho(t,x)\\ u(t,x)\end =\begin U_L, & 0U_ * , & 0\le a|x|U_R,& 0\end Although this is a simple example, it still shows the basic properties. Most important the characteristics decompose the solution into three domains. The propagation speed of these two equations is equivalent to the propagation speed of sound. The fastest characteristic defines the Courant–Friedrichs–Lewy (CFL) condition, which sets the restriction for the maximum time step in a computer simulation. Generally as more conservation equations are used, more characteristics are involved. 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 problem」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.