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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 Finite volume method ： ウィキペディア英語版 Finite volume method The finite-volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations (2002; Toro, 1999 ). Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. ==1D example== Consider a simple 1D advection problem defined by the following partial differential equation :$\backslash quad\; (1)\; \backslash qquad\; \backslash qquad\; \backslash frac+\backslash frac=0,\backslash quad\; t\backslash ge0.$ Here, $\backslash rho=\backslash rho\; \backslash left(\; x,t\; \backslash right)\; \backslash$ represents the state variable and $f=f\; \backslash left(\; \backslash rho\; \backslash left(\; x,t\; \backslash right)\; \backslash right)\; \backslash$ represents the flux or flow of $\backslash rho\; \backslash$. Conventionally, positive $f\; \backslash$ represents flow to the right while negative $f\; \backslash$ represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain, $x\; \backslash$, into ''finite volumes'' or ''cells'' with cell centres indexed as $i\; \backslash$. For a particular cell, $i\; \backslash$, we can define the ''volume average'' value of $\_i\; \backslash left(\; t\; \backslash right)\; =\; \backslash rho\; \backslash left(\; x,\; t\; \backslash right)\; \backslash$ at time $\backslash$ and $\}\; ,\; x\_\}\; \backslash right\; )\; \}\backslash$, as :$\backslash quad\; (2)\; \backslash qquad\; \backslash qquad\; \backslash bar\_i\; \backslash left(\; t\_1\; \backslash right)\; =\; \backslash frac\}\; -\; x\_\}\}\; \backslash int\_\}\}^\}\}\; \backslash rho\; \backslash left(x,t\_1\; \backslash right)\backslash ,\; dx\; ,$ and at time $\backslash$ as, :$\backslash quad\; (3)\; \backslash qquad\; \backslash qquad\; \backslash bar\_i\; \backslash left(\; t\_2\; \backslash right)\; =\; \backslash frac\}\; -\; x\_\}\}\; \backslash int\_\}\}^\}\}\; \backslash rho\; \backslash left(x,t\_2\; \backslash right)\backslash ,\; dx\; ,$ where $x\_\}\; \backslash$ and $x\_\}\; \backslash$ represent locations of the upstream and downstream faces or edges respectively of the $i^\; \backslash$ cell. Integrating equation (1) in time, we have: :$\backslash quad\; (4)\; \backslash qquad\; \backslash qquad\; \backslash rho\; \backslash left(\; x,\; t\_2\; \backslash right)\; =\; \backslash rho\; \backslash left(\; x,\; t\_1\; \backslash right)\; -\; \backslash int\_^\; f\_x\; \backslash left(\; x,t\; \backslash right)\backslash ,\; dt,$ where $f\_x=\backslash frac$. To obtain the volume average of $\backslash rho\backslash left(x,t\backslash right)$ at time $t=t\_\; \backslash$, we integrate $\backslash rho\backslash left(x,t\_2\; \backslash right)$ over the cell volume, $\backslash left(x\_\}\; ,\; x\_\}\; \backslash right)$ and divide the result by $\backslash Delta\; x\_i\; =\; x\_\}-x\_\}$, i.e. :$\backslash quad\; (5)\; \backslash qquad\; \backslash qquad\; \backslash bar\_\backslash left(\; t\_\backslash right)\; =\backslash frac\backslash int\_\}\}^\}\}\backslash left\backslash \}^\; f\_\; \backslash left(\; x,t\; \backslash right)\; dt\; \backslash right\backslash \}\; dx.$ We assume that $f\; \backslash$ is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension $f\_x\; \backslash triangleq\; \backslash nabla\; f$, we can apply the divergence theorem, i.e. $\backslash oint\_\backslash nabla\backslash cdot\; fdv=\backslash oint\_f\backslash ,\; dS$, and substitute for the volume integral of the divergence with the values of $f(x)\; \backslash$ evaluated at the cell surface (edges $x\_\}\; \backslash$ and $x\_\}\; \backslash$) of the finite volume as follows: :$\backslash quad\; (6)\; \backslash qquad\; \backslash qquad\; \backslash bar\_i\; \backslash left(\; t\_2\; \backslash right)\; =\; \backslash bar\_i\; \backslash left(\; t\_1\; \backslash right)$ - \frac^ f_} dt - \int_^ f_} dt \right) . where $f\_\}\; =f\; \backslash left(\; x\_\},\; t\; \backslash right)$. We can therefore derive a ''semi-discrete'' numerical scheme for the above problem with cell centres indexed as $i\backslash$, and with cell edge fluxes indexed as $i\backslash pm\backslash frac$, by differentiating (6) with respect to time to obtain: :$\backslash quad\; (7)\; \backslash qquad\; \backslash qquad\; \backslash frac\; +\; \backslash frac\; \backslash left($ f_} - f_} \right ) =0 , where values for the edge fluxes, $f\_\}$, can be reconstructed by interpolation or extrapolation of the cell averages. Equation (7) is ''exact'' for the volume averages; i.e., no approximations have been made during its derivation. This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a node. 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「Finite volume 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.