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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 MUSCL scheme ： ウィキペディア英語版 MUSCL scheme In the study of partial differential equations, the MUSCL scheme is a finite volume method that can provide highly accurate numerical solutions for a given system, even in cases where the solutions exhibit shocks, discontinuities, or large gradients. MUSCL stands for ''Monotonic Upstream-Centered Scheme for Conservation Laws'' (van Leer, 1979), and the term was introduced in a seminal paper by Bram van Leer (van Leer, 1979). In this paper he constructed the first ''high-order'', ''total variation diminishing'' (TVD) scheme where he obtained second order spatial accuracy. The idea is to replace the piecewise constant approximation of Godunov's scheme by reconstructed states, derived from cell-averaged states obtained from the previous time-step. For each cell, slope limited, reconstructed left and right states are obtained and used to calculate fluxes at the cell boundaries (edges). These fluxes can, in turn, be used as input to a ''Riemann solver'', following which the solutions are averaged and used to advance the solution in time. Alternatively, the fluxes can be used in ''Riemann-solver-free'' schemes, such as the ''Kurganov and Tadmor scheme'' outlined below. ==Linear reconstruction== We will consider the fundamentals of the MUSCL scheme by considering the following simple first-order, scalar, 1D system, which is assumed to have a wave propagating in the positive direction, :$u\_t\; +\; F\_x\backslash left(u\; \backslash right)=0.\; \backslash ,$ Where $u$ represents a state variable and $F$ represents a flux variable. The basic scheme of Godunov uses piecewise constant approximations for each cell, and results in a first-order upwind discretisation of the above problem with cell centres indexed as $i$. A semi-discrete scheme can be defined as follows, :$\backslash frac\; +\; \backslash frac\; \backslash left($ F \left( u_ \right) - F \left( u_ \right) \right ) =0. This basic scheme is not able to handle shocks or sharp discontinuities as they tend to become smeared. An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation with a step wave propagating to the right. The simulation was carried out with a mesh of 200 cells and used a 4th order Runge–Kutta time integrator (RK4). To provide higher resolution of discontinuities, Godunov's scheme can be extended to use piecewise linear approximations of each cell, which results in a ''central difference'' scheme that is ''second-order'' accurate in space. The piecewise linear approximations are obtained from :$u\; \backslash left(\; x\; \backslash right)\; =\; u\_\; +$ \frac \right)} \left( u_ - u_ \right) \qquad \forall x \in (x_, x_]. Thus, evaluating fluxes at the cell edges we get the following semi-discrete scheme :$\backslash frac\; +\; \backslash frac\; \backslash left($ F \left( u_ \right) - F \left( u_ \right) \right ) =0, where $u\_$ and $u\_$ are the piecewise approximate values of cell edge variables, i.e. :$u\_\; =\; 0.5\; \backslash left(\; u\_\; +\; u\_\; \backslash right),$ :$u\_\; =\; 0.5\; \backslash left(\; u\_\; +\; u\_\; \backslash right).$ Although the above second-order scheme provides greater accuracy for smooth solutions, it is not a total variation diminishing (TVD) scheme and introduces spurious oscillations into the solution where discontinuities or shocks are present. An example of this effect is shown in the diagram opposite, which illustrates a 1D advective equation $\backslash ,\; u\_t+u\_x=0$, with a step wave propagating to the right. This loss of accuracy is to be expected due to Godunov's theorem. The simulation was carried out with a mesh of 200 cells and used RK4 for time integration. MUSCL based numerical schemes extend the idea of using a linear piecewise approximation to each cell by using ''slope limited'' left and right extrapolated states. This results in the following high resolution, TVD discretisation scheme, :$\backslash frac\; +\; \backslash frac\; \backslash left($ F \left( u^ *_ \right) - F \left( u^ *_ \right) \right ) =0. Which, alternatively, can be written in the more succinct form, :$\backslash frac\; +\; \backslash frac\; \backslash left($ F^ *_ - F^ *_ \right ) =0. The numerical fluxes $F^$ *_ correspond to a nonlinear combination of first and second-order approximations to the continuous flux function. The symbols $u^$ *_ and $u^$ *_ represent scheme dependent functions (of the limited extrapolated cell edge variables), i.e. :$u^$ *_ = u^ *_ \left( u^L_ , u^R_ \right), u^ *_ = u^ *_ \left( u^L_ , u^R_ \right), and :$u^L\_\; =\; u\_i\; +\; 0.5\; \backslash phi\; \backslash left(\; r\_i\; \backslash right)\; \backslash left(\; u\_\; -\; u\_\; \backslash right),$ u^R_ = u_ - 0.5 \phi \left( r_ \right) \left( u_ - u_ \right), :$u^L\_\; =\; u\_\; +\; 0.5\; \backslash phi\; \backslash left(\; r\_\; \backslash right)\; \backslash left(\; u\_i\; -\; u\_\; \backslash right),$ u^R_ = u_i - 0.5 \phi \left( r_i \right) \left( u_ - u_i \right), :$r\_\; =\; \backslash frac\; -\; u\_i\}.$ The function $\backslash phi\; \backslash left(\; r\_i\; \backslash right)$ is a limiter function that limits the slope of the piecewise approximations to ensure the solution is TVD, thereby avoiding the spurious oscillations that would otherwise occur around discontinuities or shocks - see Flux limiter section. The limiter is equal to zero when $r\; \backslash le\; 0$ and is equal to unity when $r\; =\; 1$. Thus, the accuracy of a TVD discretization degrades to first order at local extrema, but tends to second order over smooth parts of the domain. The algorithm is straight forward to implement. Once a suitable scheme for $F^$ *_ has been chosen, such as the ''Kurganov and Tadmor scheme'' (see below), the solution can proceed using standard numerical integration techniques. 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「MUSCL 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.